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Pattern formation in reaction-diffusion models far from the Turing regime Kolokolnikov, Theodore 2004

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P A T T E R N F O R M A T I O N IN REACTION-DIFFUSION MODELS FAR F R O M T H E T U R I N G R E G I M E by T H E O D O R E K O L O K O L N I K O V B.Math., University of Waterloo, 1997 M.Sci., University of British Columbia, 1999 A THESIS S U B M I T T E D IN F U L F I L L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Mathematics Institute of Applied Mathematics We accept this thesis as conforming to the reciuiriecLstaifeard T H E . U N I V E R S I T Y OF BRITISH C O L U M B I A March 2004 © Theodore Kolokolnikov, 2004 A b s t r a c t In this thesis we analyse three different reaction-diffusion models These are: the Gray-Scott model of an irreversable chemical reaction, the Gierer-Meinhardt model for seashell patters, and the Haus model of mode-locked lasers. In the limit of small diffusivity, all three models exhibit localised spatial patterns. In one dimension, the equilibrium state typically concentrates on a discrete number of points. In two dimensions, the solution may consist of stripes, spots, domain-filling curves, or any combination of these. We study the regime where such structures are very far from the spatially homogenous solution. As such, the classical Turing analysis of small perturbations of homogenous state is not applicable. Instead, we study perturbations from the localised spike-type solutions. In one dimension, the following instbailities are analysed: an overcrowding instability, whereby some of the spikes are annihilated if the initial state contains too many spikes; undercrowding (or splitting) instability, whereby a new spike may appear by the process of splitting of a spike into two; an oscillatory height instability whereby the spike height oscillates with period of O(l) in time; and an oscillatory drift instability where the center of the spike exhibits a slow, periodic motion. Explicit thresholds on the parameters are derived for each type of instability. In two dimensions, we study spike, stripe and ring-like solutions. For stripe and ring-like solutions, the following instabilities are analysed: a splitting instability, whereby a stripe self- replicates into two parallel stripes; a breakup instability, where a stripe breaks up into spots; and a zigzag instability, whereby a stripe develops a wavy pattern in the transveral direction. For certain parameter ranges, we derive explicit instability thresholds for all three types of instability. Numerical simulations are used to confirm our analytical predictions. Further numerical simulations are performed, suggesting the existence of a regime where a stripe is stable with respect to breakup or splitting instabilities, but unstable with respect to zigzag instabilities. Based on numerics, we speculate that this leads to domain-filling patterns, and labyrinth-like patterns. For a single spike in two dimensions, we derive an ODE that governs the slow drift of its center. We reduce this problem to the study of the properties of a certain Green's function. For a specific dumbell-like domain, we obtain explicit formulas for such a Green's function using complex analysis. This in turn leads to conjecture that under certain general conditions on parameters, the equilibrium location of the spike is unique, for an arbitrary shaped domain. Finally, we consider another parameter regime, for which the exponentially weak interaction of the spike with the boundary plays a crucial role. We show that in this case there can exist a spike equilibrium solution that is located very near the boundary. Such solution is found to be unstable in the direction that is transversal to the boundary. As the effect of the boundary is increased, the interior spike locations undergo a series of destabilizing bifurcations, until all interior spike equilibria become unstable. A c k n o w l e d g m e n t s I a m v e r y g r a t e f u l t o m y s u p e r v i s o r , M i c h a e l W a r d . H i s e x p e r t g u i d a n c e , s u p p o r t a n d e n t h u - s i a s m have b e e n i n v a l u a b l e i n see ing t h i s w o r k t h r o u g h c o m p l e t i o n . I w o u l d a l so l i k e t o t h a n k o u r c o l l a b o r a t o r s , J u n c h e n g W e i a n d T h o m a s E r n e u x . C h a p t e r s 1 a n d 2 a re b a s e d o n j o i n t w o r k w i t h M i c h a e l W a r d a n d J u n c h e n g W e i . C h a p t e r s 3 a n d 4 are b a s e d o n j o i n t w o r k w i t h M i c h a e l W a r d . C h a p t e r 5 is a j o i n t w o r k w i t h T h o m a s E r n e u x . T h i s w o r k has b e e n s u p p o r t e d b y N S E R C P G S B f e l l o w s h i p a n d t h e U B C g r a d u a t e f e l l o w s h i p . i i i T a b l e o f C o n t e n t s Abstract ii Acknowledgments iii Table of Contents iv List of Figures v Introduction vii Chapter 1. Gray-Scott model in one dimension 1 1.1 Low-feed regime of the Gray-Scott model 3 1.1.1 Symmetric spike Equilibria solutions 3 •1.1.2 Non-local eigenvalue problem 6 1.1.3 Hopf bifurcation away from the saddle node point 17 1.1.4 Hopf bifurcation in the intermediate regime 20 1.2 High-feed and intermediate regime 21 1.2.1 Equilibria in high-feed and intermediate regime 22 1.2.2 Slow drift and oscillatory drift instabilities 31 1.3 Discussion 47 Chapter 2. Stripe and ring-like solutions of the Gray-Scott model in two dimen- sions 51 2.1 Ring and stripe solutions in the low-feed regime 53 2.1.1 Ring equilibria solutions 54 2.1.2 The breakup instability in the low-feed regime 59 2.2 Stripe in the Intermediate and high-feed regime 63 2.2.1 Zigzag instabilities of a stripe 64 2.2.2 Breakup Instabilities of a Stripe " 73 2.3 Stability of a Ring Solution in intermediate and high-feed regimes 77 2.3.1 Ring radius and splitting in the high-feed regime 78 2.4 Numerical Examples 88 2.5 Discussion 94 Chapter 3. Gierer-Meinhard model in two dimensions 99 3.1 Dynamics Of A One-Spike Solution 103 3.2 Limiting Cases Of The Dynamics 112 3.2.1 Dynamics For Small D 112 3.2.2 Dynamics For Large D 117 3.3 Exact Calculation Of The Modified Green's Function 119 3.3.1 Uniqueness Of The One-Spike Equilibrium Solution For Large D 125 3.4 Numerical Experiments and Discussion 129 3.4.1 Boundary Element method 130 iv 3.4.2 Experiment 1: Effect Of e Wi th D = 1 132 3.4.3 Experiment 2: Effect Of D Wi th e = 0.01 133 3.4.4 Experiment 3: Uniqueness Of Equilibria For Large D 134 3.4.5 Experiment 4: A Pitchfork Bifurcation 134 3.4.6 Discussion 137 3.5 Appendix: The Behavior Of RQ On The Boundary 138 Chapter 4. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model and near-boundary spikes 141 4.1 The dynamics of a spike for exponentially large D 144 4.2 A Radially Symmetric Domain: D Exponentially Large 146 4.3 A Dumbbell-Shaped Domain: D Exponentially Large 150 4.3.1 The Neck Region of the Dumbbell 151 4.3.2 The Lobe Region of the Dumbbell 155 4.4 Spike Equilibria Near the Boundary 157 4.5 Discussion 163 4.6 Appendix: The Proof of Proposition 5.3 165 Chapter 5. Q-switching instability in passively mode-locked lasers 168 5.1 Stability analysis ,. 170 5.2 Hopf bifurcation for for p = 3, k(t) = ^ 174 5.3 Hopf bifurcation without saturation 175 5.4 Discussion 177 Bibliography 179 v L i s t o f F i g u r e s 1 Example of Turing instability xii 1.1 Example of an equilibrium two-spike solution to 2 1.2 Undercrowding effect and oscillatory drift in Gray-Scott model 3 1.3 The function / , given by (1.21), for real values of A 11 1.4 Large eigenvalue as a function of r 14 1.5 Plot of vm = _(0) versus t 19 1.6 Fold point in high-feed regime 26 1.7 The graph of the dimple eigenfunction 32 1.8 Hopf bifurcation threshold 42 1.9 Example of oscillatory drift with one spike 45 1.10 Example of oscillatory drift with two spikes 46 1.11 Bifurcation diagram of symmetric and asymmetric spikes in the low-feed regime . . . . 48 2.1 Three different instabilities of a stripe 52 2.2 ' Labyrinth-type pattern in the Gray-Scott model 53 2.3 Ring radius vs. A 58 2.4 Domain half-length I versus the first unstable mode of a zigzag instability 66 2.5 Plot of the re-stablization mode versus B 72 2.6 Plot of instability band as a function of B 74 2.7 The graph of the first unstable mode m/ versus the ring radius 83 2.8 Ring breakup in 2D, A = 2.0 89 2.9 Ring breakup in 2D, A — 2.5 90 2.10 Ring breakup in 2D, A = 4.0 ' 90 2.11 Example of zigzag instability 92 2.12 Eigenfunction corresponding to the zigzag instability 93 2.13 Example of breakup instability : 94 2.14 Example of a zigzag instability of a stripe in the absence of breakup instability 95 2.15 Cross-section of a zigzag pattern at different times 95 2.16 Example of a formation of a labyrinthine pattern 96 3.1 A spike solution for the Gierer-Meinhardt system in 2D 100 3.2 A dumbell-shaped domain and spike equilibrium location 102 3.3 Metastable motion of a spike 114 3.4 A family of dumbell-shaped domains 126 3.5 Numerical solution using the moving mesh method 132 3.6 Spike drift as a function of e of a spike 133 3.7 Spike drift as a function of D 134 3.8 Plot of V i ? m o for a dumbell-shaped domain 135 3.9 Subcritical bifurcation of a equilibrium location 135 3.10 Bifurcation diagram for different domain neck widths 136 3.11 Bifurcation threshold as a function of domain neck width 137 4.1 equilibrium bifurcation diagram for different values of D 142 vi 4.2 Near-boundary spike locations and stability 143 4.3 Bifurcation diagram in the near-shadow regime 149 4.4 Dumbell-shaped domain and the largest inscribed circle 155 4.5 Plot of dxRmo(x, 0) versus x 156 5.1 Hopf bifurcation in the Haus model 175 5.2 Nonlinear effects of an oscillatory spike 177 vii I n t r o d u c t i o n Reaction-diffusion systems are multi-component models involving diffusion and non-linear in- teraction among the components. Such systems are commonplace in many areas of physics, chemistry and biology. They are used as a model for such diverse phenomena as cell differen- tiation, reaction of chemicals, propogation of flame fronts, laser interference patterns and sea shell patterns. In this thesis we study three different reaction-diffusion models. The first model, studied in §1 and §2, is the Gray-Scott system [28], [29], initially formulated with no diffusion. It models an irreversible reaction involving two reactants in a gel reactor, where the reactor is maintained in contact with a reservoir of one of the two chemicals in the reaction. Wi th diffusion, in dimensionless units it can be written as VT = DVAV - (F + k)V + UV2 in fl, UT = DVAU + F(l -U)- UV2 in ft, (1) ® = £ = 0 on dfi , where Vl is a bounded domain; the unknowns U(X,T), V(X,T) represent the concentrations of the two biochemicals; Du, Dy are the diffusion coefficients of U and V respectively; F denotes the rate at which U is fed from the reservoir into the reactor, and A; is a reaction-time constant. This model is known to have a rich solution structure for various parameter ranges. For example spikes, travelling waves, self-replication of pulses, oscillating spikes, and spatio-temporal chaos have all been observed. See [28], [29], [52], [53] for experimental work. For numerical and analytical results, see for instance [18], [20], [35], [53], [52], [57] [58], [59], [63], [64], [65], [66], [67], [79], [80], [85]. The second model, studied in §3 and §4, is the Gierer-Meinhardt system. Introduced in [54], it is used to model various localization processes including biological morphogenesis and sea-shell patterns (cf. [26], [54], [33]). In dimensionless form, it can be written as: At = e2AA-A+ — , xetl, t>0, (2a) rHt — DAH — H + , x e SI, t>0, (2b) dnA = dnH = 0 , xedCl. (2c) Here Cl is a bounded domain, A and H represent the activator and the inhibitor concentrations, e 2 and D represent the diffusivity of the activator and inhibitor, r is the inhibitor time constant, dn denotes the outward normal derivative, and the exponents (p, q, r, s) satisfy p > 1, q > 0, r > 0, s > 0, - - ^ < . (3) q s+1 The last model, studied in §5, is the continious limit of the Haus master equation (see [23] [31], vii i [41]), which describes the operation of a mode-locked laser: NT = 7 A — N — NL~l [> \E\ r rL E-+ Egg (4a) 2d9 (4b) Jo A common feature of these models is that their solution exhibits spikes: a highly localized structure in space, which is concentrated on a discrete number of points of the domain. A n example of such a solution is shown in the lower right corner of Figure 1. Background of Gray-Sco t t and Gie re r -Me inhard t models The modern theory of pattern formation begins with the seminal 1952 paper by Turing [71], which uses the linear analysis to determine threshold conditions for the instabiloity of spa- tially homogeneous equilibria of general two-component reaction-diffusion systems. However Turing's method is limited to patterns which are near-homogenous in space. As such, it fails to predict the stability and dynamics of spike-type solutions which are ubiquitous in many reaction-diffusion systems. The stability and dynamics of spike solutions - where Turing's approach is not applicable - is the main topic of this thesis. A more recent criteria for pattern formation was proposed by Gierer and Meinhardt [26], [43] and independantly by Segel and Jackson [68]. They postulate that the following two conditions are essential for pattern formation: local self-enhancement and long-range inhibition. Let us illustrate how these two conditions are built into Gierer-Meinhardt (2) and Gray-Scott models For the Gierer-Meinhardt model, the local self-enhancement of the activator A occurs in the regions where Ap/Hq is sufficently large (i.e. bigger than A). However A cannot increase indefinitely, since eventually the term Ar/Hs will become large, which will induce an increase in the inhibitor H, which in its turn will cancel the destabilizing effect of the Ap/Hq term. Thus the chemical H acts as a long-range inhibitor. For this reason, models of G M type are often referred to as an activator-inhibitor system. Note also that the chemical A acts as an autocatalysist, so that the activator is autocalytic. By contrast, The Gray-Scott model is referred to as an activator-substrate system. Here, the chemical U can be interpreted as a substrate depleted by V. The long-range inhibition here is due to a depletion of U. Instead of autocatalysis as in G M model, activation here is due to the presence of the source term in the equation for U. For the G M model, we suppose that the ratio £2/D is small. This implies that H diffuses much slower than A; and so near the region of local self-enhancement, one can assume that H is constant; It then follows from an O D E analysis that A must decay exponentially outside the O(e) region of self-enhancement. This is the reason for the occurence of spikes in this model. A n analogous argument applies to the Gray-Scott model, where spike-type patterns occur provided that Dv/Du is small. Note that these assumptions are in agreement with the original work of Turing, where the instability was obtained provided that the ratio of diffusivity coefficients is sufficiently large or sufficiently small. ix History of Gierer-Meinhardt model Historically, the Gierer-Meinhardt model was first proposed in a slightly different form as a simple model of a biological morphogenesis in a fresh water polyp hydra (see [26], [43]). When the stem of a hydra is cut into two, the part that is left without a head grows one. Moreover, the new head appears on the side of the stem that was the closest to the original head. It was suggested that this process is due to the graduated distribution of the inhibitor inside the stem of the hydra. In addition, Gierer-Meinhadt model in one dimension has been proposed as a model for sea-shell patterns [54]. Many variants of this model have been proposed and studied numerically in [54]. These models involve additional chemical species, saturation effects, etc. The model (2) is the simplest model in this hierarchy. Mathematically, the first rigorous analysis of the G M model was performed in the so-called shadow regime, for which D is assumed to be very large. In such a case, the chemical H becomes slave to A and, to a large degree, the analysis becomes similar to the famous P D E problem, e 2 Au - u + up = 0, x e ft, (5) dnu = 0, x E dQ, w This problem is by now well understood. In particular, a variational structure can be found in this limit. Using such variational formulation, Gui and Wei [27] have shown that as e —> 0, the solution to (5) concentrates at a finite number of points, and moreover these points are related to a ball-packing problem: they are the centers of circles of the same size that are tangent to each other and to the boundary of the domain. Similar equilibrium state exists for the G M model with large D. However such equilibrium state was shown to be unstable in [83] and [84] as long as D > 1. A more intricate picture emerges as D is decreased. In one dimension Iron, Ward and Wei [40] derived certain thresholds D2 > D3 > . . . such that a K spike symmetric equilibrium solution is stable only if D < DK- Similar therholds were also derived in two dimensions in [77]. Moreover, instead of a ball- packing problem, the locations of spike centers are now related to certain properties of Green's functions. History of Gray-Scott model Let us mention some cornerstones in the development of the Gray-Scott model. It was originally introduced in [28], [29] to describe a reaction in a well-stirred tank. The stirring rate was assumed to be large enough to ignore the effects of diffusion, i.e. DV = 0 = D U . Even in this simple form, the O D E stability analysis reveals the presence of a Hopf bifurcation. The next major developement was due to Pearson in 1993 [65]. He examined the Gray Scott model in two dimensions, in the limit DV,DU small. Using numerical simulations he observed complex pattern formation and reported many possible patterns, including spots, self-replication and labrynthian patterns. As Pearson himself observed, it is doubtful whether the Turing analysis, including the weakly-nonlinear Turing analysis, can predict such patterns, since they appeared to be far from the homogenous state. He suggested that new techniques are needed to study such patterns. Many of these patterns, including self-replicating spots, x were detected experimentally in 1994 by Lee, McCormick, Pearson and Swinney [52], where a gel reactor was used to achieve a large diffusivity ratio Du/Dv. The numerical and laboratory experiments Pearson and Swinney et al. have spurred the de- velopment of new analytical techniques to understand the these patterns. The first analytical results were derived by Doelman and his co-workers in a series of papers starting with [20]. Their theory explains analytically certain stability properties of spike-type solutions in 1-D in what we shall call here the intermediate regime. See §1.3 for more details. Independently of Doelman's work, Muratov and Osipov have also obtained many similar results for the Gray-Scott model in this regime (see [57]- [60]). In addition, they have developed a theory for pulse-splitting in one, two and three dimensions. In [63] and [64], for a different regime of the Gray-Scott model, Nishuira and Ueyama have proposed a theoretical framework which predicts pulse splitting in one dimension. Central to their theory is a certain alignment property which he demonstrated numerically for certain parameters of the Gray Scott model. Turing's analysis Let us now review Turing's approach for the analysis of the stability of spatially homogeneous stead-state solutions to reaction-diffusion systems. This approach is the usual first step for analysing biological and chemical pattern formation in both well-estabilished models such as the Brusselator [62], the Oregonator [25], and in a viriety of other systems, such as described in Murray [61]. We consider a general reaction-diffusion system in 1-D: ut = Muxx + F(u).- Here, we assume that u is a vector with n components, and M is a matrix of diffusion coefficients. Consider the homogenous solution state UQ satisfying F(uo) = 0, and consider a small perturbation from this state u = u0 + ext(f>(x), \<f>\ < 1. We then obtain the following linearized system: Ac/> = M(f>xx + V-F(u 0 ) . • Substituting <j> = t>emix we then obtain the following n dimensional eigenvalue problem: \ v = [-m2.M + VF(u0)]v. In particular, we see that when M is a positive definite matrix, A < 0 whenever m is large enough. Similarly, if VF(uo) is a negative definite matrix, A < 0 when m is near zero. In between these two extremes, there may exist modes m for which the sign of A is positive. Such modes correspond to unstable sinusoidal-type perturbations of the homogenous state. When m is of O( l ) , the sign of A depends on the balance of the diffusion and the nonlinear terms. xi t=0. t=5.0 14^ 12 10 8 6 4 2 t=15.0 Li 12 10 8 6 4 2 t=1000.0 1 Figure 1: Example of Turing instability. Initial conditions at t = 0 consist of the homogenous steady state, with very small random perturbations. After some transient period, the solution converges to spike-type steady state, shown here at t — 1000. Note that the final state is very far from the initial homogenous equilibrium. Here, A — 2.3, e = 0.03, D = 1,1 = 2"/r. Let us now perform Turing's analysis on the one-dimensional Gray-Scott model. We first transform the system (1). Set (F + k-y F ' F + k' T = F + k (6) F + k ' V(X, T) = VFv(x, t), U(x, t) = u(x, t). It is easy to see that (1) is equivalent to the following vt = e2Av — v + Av2u rut'= DAu — u + 1 — v2u dnv. — 0 = dnu, x efl x e dfi. (7) First we find the spatially homogenous steady state solution (v0,u0). It satisfies: VQ = VQUQA, U0 = 1 - VQUQ. xi i Depending on the parameter A, there are either one or three solutions. A trivial solution is uo = 0, _o = 1. Two other solutions, which exist when A > 2, satisfy VQUQA = 1 , VQ - Av0 + 1 = 0 , so that A ± VA2 - 4 + 1 Vn = , U, A AVQ Next, we linearize around these equilibirium states. We let v = vo + (peXt cos(mx), u = UQ + rjeXt cos(mx). To satisfy the Neumann boundary condition, we require that ml = jir where j is a non-negative integer. We then obtain: Ac/> = -£2m2<j> - </> + 2u0v0A4> + v2Arj, T\T] = —Dm2r] — T] — 2uQVQ(f> — v2r\. This can be written as Aw = Mv where v = (c/>, ry)' and M = e2m2 - 1 + 2UQVQA v^A 2u0v0 -±(Dm2 + l + v2) First, we consider the trivial solution VQ — 0, u 0 = 1. Then the eigenvalues of M become —e2m2 - 1,— \{Dm2 + 1). Thus both eigenvalues are negative and so the trivial solution is stable with respect to all modes m. Next, we consider VQ — v^. Then we obtain: M = -e2m2 + 1 v2A - 1 1 T A Note that t r M = 1 — e2m2 - \{Dm2 + ^ o ) . There are two cases to consider. First, suppose that t r M < 0 for all m. This occurs provided that In additon we have: f(m) = d e t M = ^ {e2DmA + m2 ((1 + v2)e2 - D) + v20 - 1} . Assuming the trace is negative, we will have instability whenever f(m) < 0. Next, we calculate: T T T Since A > 2, it follows that /(0) > 0 for VQ and /(0) < 0 for VQ . We conclude that the zero mode is unstable for VQ . Thus the VQ solution does not generate heterogenous patterns. On the other hand, when e -C D and m = O( l ) , we have f(m) ~ —Dm2 + /(0). It follows that for xiii VQ, f will become negative at the point when m 2 = ^p- > 0. This shows the existence of the instability band for VQ. Since f(m) —> oo as m —» oo, we know that all modes m are stable when m is large enough. More precisely, when m » 1 we have f{m) ~ e2Dm4 - m2D, so the upper bound for the instability band satisfies: 1 £ Finally, note from (6) that we must have r > 1. In particular, since A > 2, the condition (8) is satisfied for the VQ solution whenever r € [1,2]. The second case, when t r M > 0, occurs when r is sufficiently large. In this case there is always an unstable eigenvalue. In addition, if f(m) > 0 then these eigenvalues are purely imaginary when t r M = 0. Thus a Hopf bifurcation occurs when r is increased past a certain threshold. The corresponding perturbations then oscillate in time. To conclude, Turing's analysis yields that VQ is unstable with respect to an instability band m i < m < m-2 where mi = i £ whereas v0 is unstable with respect to the zero mode. The trivial homogenous state is stable with respect to all modes. Above we have shown the Turing instability under the assumption that D is large. It is easy to show that the instability band shrinks as D is decreased until it dissapears at some therhold value of D = DT- For values of D near this threshold, it is possible to perform the weakly non- linear analysis, by expanding D and m near DT- This yields the so-called amplitude equations, which are useful to analyse the stability and other properties of the sinusoidal patterns. There is a large body of literature devoted to this topic; see for example [62], [13] for comprehensive reviews. While Turing's analysis above shows that the Gray-Scott model exhibits patterns, it says noth- ing about the final state of the destabilized homogenous state. In fact, in the regime e <C D, the final equilibrium state is very far from the homogenous state - see Figure 1. Thus Turing's analysis is insufficient to explain the final equilibrium state. The goal of this thesis is to study these far from homogenous state solutions. M a i n contr ibut ions and compar ison to previous work There has been many related works dealing with localised structures in Gray-Scott system - see Section 1.3 for references. The main difference between these results and the results of Chapter 1 is that the previous works analyse a single spike on the unbounded domain. B y contrast, we study K spikes on a bounded or unbounded domains. This introduces several new phenomena which have no counterpart with previous work on the Gray-Scott model. xiv The first novel result is a simple analytic formula which predicts the minimum number of pulse-splitting events of the Gray Scott model in the high feed regime A = 0(1) on a bounded domain. The result is given in Proposition 1.2.1. While the pulse-splitting behaviour has been studied previously on an unbounded domain (see for example [57], [66]), there is no analogue of that formula there, since the boundedness of the domain is essential. Another new result for the Gray-Scott model is the the overcrowding instability thersholds of multi-spike solutions, as given in Proposition 1.1.7. This result is analogous to the thresholds computed previously for the Gierer-Meinhardt model (see [40]), and again, the boundedness of the domain and allowing for more than one spike solution is essential here. Finally, the boundedness of the domain is crucial in the analysis of oscillatory travelling spikes. A travelling spike instability was predicted for the infinite domain in [60], where a specific threshold was also derived. However the oscillation in the position of the center of the spike can only occur in the presence of a bounded domain. This oscillation is due to a new type of Hopf bifurcation that is analysed in §1.2.2 The main result of Chapter 2 is the analysis of stripes of the Gray-Scott model in two dimensions. In particular, the technique used to study the zigzag instabilities of Section 2.2.1 is new. The main contribution of Chapter 3 is an explicit computation of the regular part of the reduced wave Green's function with Neumann boundary condition. We then make a conjecture about uniqueness of the maxima of such a function. The importance and novelty of this conjecture is described in Section 3.4.6. In Chapter 4, one of the main results is the existence and stability analysis of a near-boundary spike equilibria that occurs in the case when the diffusivity coefficient D of the Gierer-Meinhardt is is exponentially large. The main result of Chapter 5 is the analysis of a Hopf bifurcation in the model (4) of mode- locked lasers. This provides an explanation of oscillations that were numerically found in in [41]. Thesis out l ine The outline of this thesis is as follows. In §1 we study the stability of spike solutions in the the one-dimensional Gray-Scott model. We consider four different types of instability that may occur. The first type, analysed in §1.1.2, is an overcrowding or competition instability. As a result of such an instability, some of the spikes are annihilated if the initial solution profile consists of too many spikes. We derive explicit thresholds on the maximum number of spikes that are stable with respect to such an instability. This number is found to be an increasing function of the domain length. The second type of instability, studied in §1.1.3 and §1.1.4, results in an oscillatory behaviour of a spike, whereby its height oscillates periodically in time. The third type is an undercrowding or splitting instability. In the presence of such an instability, successive spike replication is observed, until a certain threshold number of the spikes is exceeded. This threshold is found to be an increasing function of domain length. Explicit formulas for this number are given in §1.2.1. In §1.2.2, we study an oscillatory spike drift, whereby the location of the center of the spike oscillates on a slow time scale. In §1.3 we compare our work to previously known results, in particular the work of Doelman and collaborators ([16]-[21], [55]) and the work of Muratov and Osipov [57]-[60]. The results of this chapter have been previously reported in [74] and [44]. xv In §2 we consider ring-like and stripe-like solutions of the Gray-Scott model in two dimensions. Such solutions are one-dimensional in nature, and many of the techniques from §1 are applicable with some modifications. We focus on two types of instabilities: breakup and zigzag instabilities. When a stripe is unstable with respect to a breakup instability of mode m, it breaks up into m radially symmetric spikes. A zigzag instability on the other hand deforms a stripe into a zigzag- like pattern. These instabilities are illustrated in Figure 2.1. We characterize the instability band for both types of instabilities, and for either a ring or a stripe. In the parameter regime where analysis is possible, we find that the stripe is always unstable with respect to some band of breakup instabilities; and may or may not be stable with respect to zigzag instabilities. We also numerically find a regime where the stripe does not break up but is unstable with respect to zigzag instabilities. We conjecture that this behaviour leads to labrynthian patterns, such as shown on Figure 2.2. The results of §2 were previously reported in [45]. In §3 and §4 we study the slow drift of a single spike solution of the Gierer-Meinhardt model in two dimensions. There are two effects that contribute to this drift: the first is the effect of the boundary, which is typically exponentially small in the distance to the boundary. The second is due to the inhibitor and depends on the diffusivity constant D. In §3 we first consider the latter effect, ignoring the exponentially weak effect of the boundary. For D small with 0(e1) <C D <C Oil), the spike motion is found to be metastable, with stable equilibrium positions located at points furthest away from the boundary. Thus for a dumbell shaped domain such as shown in Figure 3.2, the equilibria would be located at the centers of lobes of the dumbell. On the other hand for D 3> 0(ln i ) , the motion depends on the gradient of a modified Green's function for the Laplacian. For a certain class of dumbell-shaped domains, we use complex variables to derive an explicit expression for such a function. Using this formula, we then show that the equilibrium is unique and it is located in the neck of the dumbell. This leads to a general conjecture 3.3.3 about the modified Green's function, that states that its regular part has a unique minimum in the interior of an arbitrary shaped simply-connected domain. Aside our analytical example, several numerical examples are also investigated, all supporting this conjecture. The results of this section were previously presented in [46].. In §4 we consider the limiting case of exponentially large D, when the effect of the boundary cannot be ignored. In the limit D = co the Gierer-Meinhardt reduces to a well known shadow problem (see for instance [37]) for which it is known that a single interior spike will drift exponentially slowly towards the nearest boundary. This is in contrast to the dynamics when when D is large enough (but not not exponentially large), in which case a single interior stable equilibrium position exists, as explained in §3. Thus an intricate bifurcation structure occurs in the regime where D is exponentially large. In §4.3 this structure is analysed for a family of dumbell-shaped domains that were introduced in §3. This transition, from a stable interior spike when D is large, to the dynamics driven by the nearest boundary in the case D = co, is made possible by the existence of what we call near boundary spikes, in the transition regime when D is exponentially large. They occur at a distance a from the boundary, where e < f f < l and D ~ c ( ^ Y / 2 e ^ , where C is some positive constant. For our dumbell-shaped domain, we find that their stability is as shown in Figure 4.2. The near-boundary spikes are always unstable in the the direction normal to the boundary. Their stability in the tangential direction depends solely on the xvi properties of the modified Green's function on the boundary. The results of this section were previously presented in [47]. In the last Chapter 5, we apply some of the techniques of §1 to the the model (4) of mode- locked lasers. In [41], this model was studied numerically. It was found numerically that as the parameter a is increased, the spike starts to exhibit periodic oscillations in height. We show that this behaviour is due to a Hopf bifurcation of the linearized problem. Moreover we are able to accurately predict the threshold value at which this bifurcation occurs. Our theoretical predictions compare favorably with numerical simulations. The results of this section were previously presented in [23]. xvii C h a p t e r 1 Gray-Scott model in one dimension In this chapter, we consider the one dimensional version of the Gray Scott model (7), Vt = £2VXX — v + Av2u , xe[-l,l] TUt = Duxx — u + 1 — v2u (1-1) vx{±l,t) = 0 = ux(±l,t). We are concerned with the regime when the ratio of diffusivities is small, £ « 1 , £ 2 « D. A typical solution is shown on Figure 1.1. The distribution of the chemical v typically consists of k localized spikes, concentrated at certain points of the domain, with negligible amount of v away from these points. The concentration of u vary over the length of the domain. In one dimension, as we will show, there are three important regimes, depending on the param- eter A: • . Low-feed regime: A = 0(e1^2) (1.2) Intermediate regime: 0 ( £2 ) <^A<^0(1) (1.3) High-feed regime: A = 0(1). (1.4) The main difference between the low-feed and high-feed regime is that in the low-feed regime, the chemical u changes on a much slower scale than v, both near the center of the spike, and in the region away from the center. Thus (1.1) is weakly coupled in the low-feed regime. On the Figure 1.1: Example of an equilibrium two-spike solution to (1.1), in the high-feed regime. Here, D = 1, e = 0.03,1 = 2, A = 1.7, r = 1. other hand, in high-feed regime, u and v are strongly coupled near the core of the spike, but are weakly coupled away from the core. The intermediate region can be viewed as a limit case of between high-feed or low-feed regimes. Depending on the regime, several different types of instability may occur. In §1.1 we analyse the low-feed regime. There, we study two types of instabilities. First, there exists an overcrowding instability: there are thresholds fa < fa... such that if I < fa then k spikes located symmetrically on the interval [—I, I] are unstable. As a result, some spikes will quickly dissapear until there are k' spikes left, where k' is such that l> fa>. Second, we show the existence of an oscillatory profile instability. It is manifested by the periodic oscillations of the spike height, when the parameter r is large enough. In §1.2 we study the intermediate and high-feed regimes. In these regimes, we study two addi- tional types of instability. We show that in the high-feed regime, an undercrowding instability will occur whenever the constant A t a n h ^ ^ is above 1.35. This type of instability leads to spike splitting, whereby one spike self-replicates into two spikes. A n example of this is shown in 2 Figure 1.2(a). In either the intermediate or the high-feed regime, an oscillatory drift instability occurs if r is sufficiently large. This type of instability results in a slow periodic motion of the centers of the spikes. A n example is given on Figure 1.2(b). Finally, as in the low-feed regime, periodic 0(1) oscillations of spike height may also occur. In Proposition 1.2.10 we derive a scaling law which predicts whether oscillatory drift or oscillatory profile instability occurs first. We conclude with §1.3 where we compare the results of this chapter with the existing literature on the Gray Scott model. xo o xO cH A "*\ A A 'J V, •J V -J -J V A. A. -' ^ *"—•*' : / ; \ ;' '•J j \ f\ f\ /\ A '• / / : / \ / /' '•- 300 200 400 600 800 1000 1200 1400 1600 1800 2000 time (a) (b) Figure 1.2: (a) Undercrowding effect in Gray-Scott model: Positions of spike's centers are plotted vs. time. Here, 1 = 2, D = 1, A = 3.5, e = 0.03 and r = 1. (b) Oscillatory drift: two spikes in asynchonious, periodic drift. Here, 1 = 2, D = 1, e = 0.03, A = 1.64, r = 88. 1.1 Low-feed regime of the Gray-Sco t t m o d e l 1.1.1 Symmetric spike Equilibria solutions We begin by asymptotically calculating the equilibrium spike solution of (1.1). It is convenient to introduce the following scaling: A = el'2A, v = e~l'2v. The system (1.1) then becomes: 3 vt = £ vxx - v + Av1u rut = Duxx — u + 1 — \v2u isx(±l,t) = 0 = ux(±l,t) , xe[-l,l] (1.5) We seek the equilibrium solution in the form of symmetric k spikes, whose centers are located at 2i — Is Xi=l[-1 + — — • ) , i = l....k. k (1.6) Near the core of each spike, we write . X Xi \ ( X Xi v ~ £w [ — - — I , u ~ U where w,£,U are to be found. We then obtain: w" - w + w2£UA = 0, U" - £2U + e2 - £i2w2U = 0. Thus, to leading order we obtain: U ~ u(x{), and choose £ = 3^7. Then w(y) satisfies w" - w + w2 = 0, u / (0 )=0 , w(y) ~ C e ~ | y | as -> 00. The explicit solution of (1.7a) is u> = - sech2 ( - 2 V 2 / To find (7, we perform asymptotic matching with the outer solution. We have 1 fl u(x) = 1 - - / G(x, x')u(x')v2(x') dx', £ J-i where G is the Green's function satisfying DGXX-G=-5(x-x'), Gx(±l,x') = 0. Changing variables, noting that U(xi) = U and £ = ^ we thus obtain, k IG(XJ,X')U(X')V2(X) dx' ~ / G(xj,Xi + £y')u(xi + £y')u2{xi + ey') dy' i=l ^ fc ^Y^G(xjtXi)U^ Jw2 (1.7a) (1.7b) 1 i=l 6CT t/,42' where k a = J^G(xj,Xi), (1.8) 2 = 1 and we have used r w2 = 6. (1.9) Here and below, the integration is assumed over entire space unless limits are specified. It remains to evaluate a. Rather than doing this directly, we define u(x) = Y J i G(x, Xi). Then a = U(XJ). Notice that u satisfies k Du"-u =-^T5(x-Xi), -l<x<l; u'(±l) = 0. t=i This problem is equivalent to: Du"-u = 0, Du(xf) -Du'(x-) = - 1 , u'(±Z) = 0. By symmetry, we obtain: Du'(x^) = ^,Du'(xf) = —^. A simple calculation then yields cj = ^ c o t h A , where c?0 = - U (1.10) 2 k AV Y) From above we have U(XJ) ~ 1 — JJ^- Since U = U{XJ), we get that U satisfies the quadratic equation The solution of this equation, U = l - Q o - ^ . (1.11a) 1 / 12t90coth^a ^ = 5 " V 1 — ( l l l b ) exists, with [/_ € (0,1/2], (7+ 6 [1/2,1), if and only \i A> Ake where Ake = Z ) - 1 / V l 2 c o t h / t 9 o A > #o = - U (1-12) We summarize our results as follows. Proposition 1.1.1 Suppose that A > Ake, with Ake given by (1.12). Then there exist two symmetric k-spike equilibria solutions to '(1.5). They are given by k where^Xi is given by (1-6), w is given by (1.7) and U± is given by (1.11). We shall denote (u+,v+) the large solution, and (u~,v~) the small solution. 1.1.2 Non- loca l eigenvalue prob lem Next, we derive an eigenvalue problem that characterizes the stability of of the symmetric fc-spike solution of (1.5), with respect to perturbations that are even functions in the core of each spike. We shall call the corresponding eigenvalues large eigenvalues. This is to distinguish them from the small eigenvalues which arise from near translation invariance. The stability with respect to small eigenvalues will be discussed in §1.2.2 below. The large eigenvalues are so called because they remain O( l ) as e —> 0. By contrast, the small eigenvalues tend to zero as e —> 0. We start by linearizing around the equilibrium state. We let v = z/±(x) + eXt4>{x), u = u±(x) + .e^xp(x). The linearized system then becomes Ac* = £24>" - <j> + 2Av±u±<f> + Av±ip, (1.13) rXtp = Dip" ~ ^ - \ {2v±u±(f) + v\\if) . (1.14) Near Xj, we have e — - r + , V>~^- (1-15) Here, the constants Cj,ipj,j = 1,... k are to be found later by asymptotic matching of the outer and inner solution. We first determine the effect of the inner solution. Near x = Xj we then have using Proposition 1.1.1 that -V-j-U-j-6 ~ TVJ& ~ &(x — Xj)-^r f £ £ A A J 1 2 ; iv \ ^3 -V±1p ~ d{X — Xj)- 1A2Ul Thus we have: D 6 The solution may be written as: DA DA2Ul where tf> = [ipi,...il>k]t, c= [ci,..., CkY, g = [G(xi,Xj)]. Here G{x, y) is the Green's function satisfying Gxx-62xG = -5(x-y), Gx(±l,y) = 0. This yields: DA2Ul DA Substituting into (1.13) we obtain the nonlocal eigenvalue problem A c * = c L 0 * - Mc !w2 ' where and the matrix M is defined as L 0 $ = + 2u;*, (1.16) M = ^Jl^ (6 + — U r s l ^ . UlDA2 V J i t ; * It follows that J ™ where x is an eigenvalue of M ' , with c* its corresponding eigenvector. It remains to find the eigenvalues of M = M*' . First, we claim that g 1 is the k by /c tridiagonal matrix - l + 2c - 1 - 1 2c - 1 _ X _ 0 X - 1 2c - 1 - 1 2c - 1 - 1 - l + 2c (1.17a) 7 where 21 s = smh(6xd), c = cosh{6xd), = —. (1.17b) To see this, we consider the system QX = B, i.e. J2j G{xi:Xj)Xj = Bi. But then Bj = U(XJ) where u satisfies u"(x) - e\u(x) = - ^ XjSixj - a;) with u'(±l) = 0. So for 1 < i < K we have: Bi+i = ^-u'(xf)s + BiC, u'(x7+i) = u'(xt)c + Q\Bis, Xi = u'(x7) - u'(xf). It follows that: u'(xf) = ^(Bi+1-Bic), Xi = u'(x tti)c + OxBi-is - u'(x?) = — {Bi - Bi-ic) c + OxBi_lS - ^ - BiC) s s = ^ ( - B i . ! + 2BlC - Bi+l). s For i = 1 we have, on the interval [—l,x±], u = acos[{x + l)6\] where a is some constant. Thus: Bi = a cosh —6\, Li u'(x7) = a0 A s inh^0 A , d \ t T « a sinh - ^ ( B 2 - B , c ) , X i = ct#A sinh -^x-~ u'(xf c o s h ( ^ A ) 2 $A s ( 5 i ( 2 c - l ) - S 2 ) , where we have used the identity sinh(f) _ cosh(x) - 1 cos(|) sinh(x) 8 This proves (1.17a). Next we note by direct computation that the eigenvalues of the k x k matrix r - l - l -1 o -1 -1 o -1 A= • -1 0 -1 -1 -1 are given by -2cos( 7 f ( j ; 1 } ) , j = l-..k k with the corresponding eigenvectors '^U ~ 1), Is . . , vmj = cos \ (m - tj) J - m = l---k It follows that the eigenvalues of Q 1 are: 0\ f0 0 vr(j - 1) Ii j = — 2c — 2 cos —̂— - s V k = 29x with the corresponding eigenvectors tanh(t9A//c) + ( 1 - cos ^ ^ ) csch(2c9A/A:) , J = 1 •' • k, Cmj = cos ^ k ^ (m - 1/2) j . Thus we obtain the eigenvalues of M , which we write in the following form: 2s Xj where We summarize. s = s + /j,jcrD U±-l u± • Proposition 1.1.2 In the limit e —> 0, the large eigenvalues of (1.13) satisfy: A $ = L0<& - x fw2 ' (1.18a) where <3>(r) = 4>(ex) L 0 $ = $ " - $ + 2w$ (1.18b) and x = Xj, j = 1.. . k given by Xj = j = l...k (1.18c) s + Y tanh(Z6>A/fc) + ( l — cos where 9 A = (1.18d) C / ± - l (1.18e) We s/iaZZ re/er io (1-18) as a non-local eigenvalue problem (NLEP). The non-local eigenvalue problem (1.18) has been extensively studied in a series of papers. In [18], a computer algebra assisted approach was developed to determine the spectrum of of (1.18). However their approach relies on manipulation of hypergeometric functions, and requires the use of computer algebra. Here, we will use a different method that does not require hypergeometric functions, developed in a series of papers [78], [77], and [74]. The basic result, shown in [78] and [77], is the following. T h e o r e m 1.1.3 (See [78]) Consider .the problem (1.18a), and suppose that x is a constant independent of X. Let X be an eigenvalue of (1.18a) with largest real part that corresponds to an even eigenfunction $. Then Re(A) < 0 i / x > l , A = 0 with $ = w if x — 1> and Re(A) > 0 * / X < l - . ^ Suppose that x = x(Tty ^s continuous and defined on ]R+. / /x(0) < 1 then Re(A) > 0. Note that the threshold case x = 1 follows from the observation that LQW = w and f w2 = f w. The proof for non-constant x is deferred until Proposition 1.1.6. For the rest of the proof, see [78], Theorem 2.12. Note also that the assumption that $ is even is necessary, since A = 0, $ = w' satisfies (1.18) for any x- 10 For the case x — 0, the following result is known: L e m m a 1.1.4 (From [50]): The problem Lo$ = A$ admits a positive eigenvalue XQ = 5/4, with the corresponding eigenfunction $ = sech 3(j//2) ; a zero eigenvalue with eigenfunction $ = w' and a negative eigenvalue A = —3/4 with the corresponding eigenfunction 5sech 2(y/2) — 4sech(y/2). We now study the case where % depends on A. This analysis first appeared for the Gierer Meinhardt model in [77], [74]. By scaling we.may write (1.18) as 10- 8- 6^ 4- 2- 0 -2 - 4 - 6 -8-] -10 Figure 1.3: The function / , given by (1.21), for real values of A = ( ^ o - A ) W . Thus the N L E P problem (1.18) is equivalent to finding the roots of ( A T ) = /(A) where / ( A ) = f W(L°~ lw2 ^ . c , x , = ]_ ^ ^ J ™l Xj Explicitly, we have: 1 1 0_x h Ci = - + - - 3 2 2s " twh{Wx/k) + [ 1 - cos ^ ^ ).csch(2Z0A/fc) /c coth(W0/k), j = 1.. . k. (1.20) The following lemma summarizes some properties of / . 11 L e m m a 1.1.5 (see [77]) The behaviour of f near the origin is: / ( A ) ~ 1 + ^-A + A C c A 2 + 0 ( A 3 ) , where KC = ^ ^ > 0. (1.21) 4 J wz In addition we have the global behaviour: / ' ( A ) > 0 , / " ( A ) > 0 , A € (0,5/4). Moreover f has a singularity at A = 5/'4 with f —> ±oo as A —> 5 /4 ± . On i/ie o£/ier side o/ i/ie singularity we have / ( A ) <0, A > 5 / 4 , and 5 / (A) ~ - — as Re(A) -> oo. The graph of / for real A is shown on Figure 1.3. Note the singularities at 5/4 and —3/4, which are the discrete eigenvalues of the LQ. The first result with Cj non-constant concerns instability. We show the following: P r o p o s i t i o n 1.1.6 (See also [77]) Let 1 / 7 r ( i - m i (1.22) 1 _ 1 Bj = — = t + — H s 1 — cos — ^ - 2s 4ssinh 2/t9 0/fc V k Suppose that Bj > 1 for some j. Then the NLEP problem (1-18) is admits a purely positive eigenvalue. Thus a k-spike equilibrium derived in Proposition 1.1.1 is unstable. In the inner region near the rrft1 spike, the perturbation of the v-component of the equilibrium solution that corresponds to j-th large eigenvalue has the following form: Kx> t] ~ {'MZ W ic^) + / 9 C m i * ( £ ~T = i ) eAt) ' 1 1 " X m l = 0 ( £ ) ( L 2 3 ) where 3 <C 1 is some small constant, $ is an even function defined in (1.15), and cmj = s i n ( ^ ( m - 1/2) V (1.24) 12 Proof. We have /(0) = 1. Thus /(0) < Bj. But / blows up at 5/4. It follows that by the intermediate value theorem that / and Cj intersect at some A € (0, 5/4). Q It is easy to see that we have the following ordering principle: B\ < B<i < ... < Bk. Therefore the first node that undergoes an instability in Proposition 1.1.6 \s Bk- The threshold case Bk = 1 corresponds to the value 1 + cos ix Ik s = sk = H — ' — (1.25) 2sinh2Zt90/A; V or equivalently, rzz\ . , 11 + COS IXIk I = lk = VDk arcsinh J —— (1.26) V ( s — •*•) Using the identity: A 1 + s Ake 2,/s ' (1.27) this threshold is also equivalent to l+cos ir/k A = Ak = Ake{ , 2 s i " h ' e ° A y _ ( 1 2 8 ) 1 ' l , l+cos7r/fc 1 + 2sinh 2;0 o/fc In addition, in view of Lemma 1.1.3 and the ordering principle, we have stability whenever Bk < 1 and r = 0. Next, note that B\ = \ + ^ . It follows that we have instability whenever s < 1. But from.(1.18e), we have s < 1 for t7 + , s > 1 for t7~. We summarize the results as follows. T h e o r e m 1.1.7 The large equilibrium solution u+,u+ given in Proposition 1.1.1 is always unstable. A one-spike negative solution is always stable provided that r = 0. The small k-spike equilibrium solution v~,u~ is unstable, provided that k > 2 and A < Ak or I < Ik, where Ak and lk are given by (1.28) and (1.26) respectively. The converse is true when r = 0. Note that the small and large /c-spike solutions are connected through a saddle-node structure, as illustrated by solid curves in Figure 1.11. For a single-spike solution, since s\ = 1, this 13 T/,=3 A = ± 0 . 2 2 i A -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (a) (b) Figure 1.4: (a) The plot of / (A) (solid curve) and Ci(A) (dashed curves) for three different values of r as indicated, with Ci(0) = 0.9. (b) The path traced by A in the complex plane for different positive values of r. The arrows indicate the direction of the path as r is increased. Note that A is negative real when r < 1.5, A is complex for r G [1.5,5.3] and A is positive real when r > 5.3. A Hopf bifurcation occurs at r = 3. theorem shows that the small solution branch is stable. For k > 1, part of the small branch near the fold point is also unstable. A consequence of this theorem is that a domain of a given length 21 can support at most k spikes where k is some increasing function of I. If the initial condition consists of more than k spikes, then numerical simulations indicated that some of them will be annihilated until at most k are left. We thus refer to such an instability as competition or overcrowding instability. The name competiton We next study the effect of r on stability. In general, if r is sufficiently large, the system can be destabilized via a Hopf bifurcation. This was first proved in [77] for the Gierer-Meinhardt model, and in [74] for the Gray-Scott model. In general, there is no explicit formulae for the threshold Th at which the Hopf bifurcation occurs (although rigorous estimates have been derived in [77]). However explicit formula is available when s is slightly above s^. For clarity, we will only treat the case of a single spike here. The existence of a Hopf bifurcation in such a case is easily seen geometrically, as shown in 14 Figure 1.4, and explained below. When s is slightly above si = 1, Ci(0) will be slightly below 1, and since / has a positive slope at the origin, it follows that for / will fall below Ci(0) when A is sufficiently negative. Note that increasing or decreasing r corresponds to compressing or expanding the curve C\(T\) along the x-axis, respectively. Moreover, C\ —> oo as A —> oo. It follows that / and C\ will intersect tangentially at some A < 0 for some r = r_ sufficiently small; and again at some A > 0 for some r = r + > r_. In between r_ and t+, the intersections occur in complex plane; thus by continuity they must intersect the imaginary axis at some TH e [ t _ , t + ] . Next we compute the precise value of T^,T±. for s near 1. Assume that s = 1 + 6, 0 < 6 < 1. After some algebra, we obtain tanh 16\ tanh WQ from which = 1 + zW0 csch 2160 - z2 (l8o\ csch 2160 + ^l26^ sech2 W^j + 0 ( z 3 ) , z = Ar, Ci(A) = 1 - ^ + / 3 z - 7 z 2 + 0(2 2 ,5 2 ,z(5) , where z = Ar, B= - + -^csch2/c9 0 , 7 = - - -Zt?0csch2Zc? + -Z 2t? 2 sech2 W0 (1.29) 4 2 8 4 8 Moreover, using some calculus, one can show that 7 always positive. It follows that the eigen- value A, given by the intersection of C\ and / , satisfies: ^ + A ( ^ - / ? r ) + K + 7 r 2 ) A 2 = 0. These curves intersect tangentially provided that in addition, This yields an equation for r , ( - - / 3 r ) + ( « c + 7 1 - ^ = 0. O = { \ - B T ) 2 / { K C + 1 T 2 ) , whose solution is 3 _ v i v ) 15 The corresponding A is negative for r_ and positive for r + . When r e [r_, r + ] , the two curves / and Ci do not intersect, and so A is comp lex-valued. In particular, it is purely imaginary precisely when r = Th = This shows the existence of the Hopf bifurcation. Next, the formula (1.27) yields: A = Aie ( l + g5 2) . Finally for the infinite domain case I = co, we have (3 = 4, 7 = g- We have shown the following: Proposition 1.1.8 Suppose that A = Aie (l + £<52^) , where 0 < < 5 « 1 . (1.30) Let 3 ^ V ^ + ^ f e ) 2 T h = 4/3 T ± = T / l /? ' where (3 and 7 are defined in (1.29) and Kc is defined in (1.21). Then the small equilibria solution (v-,u-) given by Proposition 1.1.1 with k = 1, is stable if and only if r < T^, and undergoes a Hopf bifurcation at r = r .̂ Moreover, the corresponding eigenvalue A is purely imaginary at T — T^, purely real and positive for r > r + and purely real and negative for T < t _ . For the case of an infinite domain I = 00, we have r^ = 3, T± = 3 ± V 6 ^ K C + 9/8. When r G [r^jT+J, due to the eigenvalue being complex, instability is manifested as small oscil- lations in the spike height. However when r > r + , the eigenvalue is purely real, and numerical experiments indicate that the corresponding instability results in the monotinic decrease of spike height, until the spike is annihilated in 0(1) time. Thus the range of r for which oscillatory behaviour occurs is of 0(V5) when (1.30) is satisfied. Due to the scaling of the Gray-Scott model (see (6)) r must satisfy r > 1. It is easy to show that Th is an increasing function of 6Q with = | when 9Q = 0 and = 3 when 6Q = co. Thus the single spike solution that is near the threshold A\e is always stable when r = 1. 16 1.1.3 H o p f b i f u r c a t i o n a w a y f r o m t h e s add le n o d e p o i n t In the previous section we have studied the occurence of the Hopf bifurcation using local analysis near the saddle node bifurcation point. Here, we study the more general case. We proceed as in [77] and [74]. The main result is the following. T h e o r e m 1.1.9 Consider the k-spike small solution (z/_,«_) given by Proposition 1.1.1. Then for each 1 < j < k, there exists a value Th > 0 and r + > Th such that the j-th large eigenvalue of (1.18) is stable for r just below Th, and unstable for all r > r^. At r = Th, this eigenvalue undergoes a Hopf bifurcation so that (1.18) admits purely imaginary eigenvalues. For r € [ 7 7 ^ X 4 - ] there are complex eigenvalues which merge with the positive real axis at r = r + . For T > T+, there are purely real, positive eigenvalues. We first show the existence of T ^ . We substitute A = iA; into (1.18) and collect real and imaginary parts. We obtain: C{T\i) = C(ir\i), / (Ai ) = f(i\i), C = Cr + iCi, f.= fr + ifi, and C = Cj (1.31b) with Cj given by (1.20). The key properties of these functions, derived in [77] and [74], are summarized below. L e m m a 1.1.10 (see [77] and Proposition 3.7, [74]) The functions /,/- satisfy C r ( rAj ) = / r (Aj ) , Ci(rAi) = fi(Xi) where (1.31a) fr(K) = fwL0[Ll + X2} fw2 / . (Ai) = KfwjLl + Xf}-^ fw2 They have the following asymptotic behaviour: Here Kc is given in (1.21). Moreover, the functions / r (Aj) and /j(Aj) have the following global behaviour: / ; ( A i ) < 0 , / ( A i ) > 0 for Xi > 0. 17 For Xi > 0 and r > 0, the functions Cr and Ci given by (1.31) satisfy: Cr(0) = B„ C'r{TXi) > 0; Ci(0) = 0, C ^ r A , ) > 0, ^-Cr{rXi) = 0{rll2), A Q ( T A 1 ) = O ^ 1 / 2 ) as r ^ oo, a\i a\i Cr(\iT) = B3 + 0(rXi), Ci{XiT) = 0(rXi) as r -* 0. ffere, Bj — Cj(0) whose properties are given in (1.22). We now return to proof of Theorem 1.1.9. Let g = C-f, g = C-f, so that the number of unstable eigenvalues M of the mode j of (1.18) corresponds to the number of zeroes of g in the positive half-plane. To determine M , we use the argument principle. Consider a counterclockwise contour composed of the imaginary axis [—iR, iR] and the semi- circle ReL9,9 e [ - | , f ] . From (1.20) we have C ~ 0(v / A) as A r -» oo. Moreover / -> 0 as A r —* oo. It follows that the change in argument of g over the semi-circle is | as R —> oo. In addition, the function # has a simple pole at the eigenvalue 5/4 of the local operator LQ. On the imaginary axis, we use g(X) = g(X). We thus obtain the following formula for the number of zeroes M of g in the right half-plane: M = \ + ^ A [ l o o , 0 ] 5 where A[ i o oo]9' denotes the change in argument of g along the semi-infite imaginary axis [0,zoo] traversed in the downwards direction. From Lemma 1.1.10 we have gr(0) < 0, g'r > 0 and gr —> oo. It follows that gr has a unique root A* e / r - 1 ([C(0),l])> for any given r > 0. Next, note that g ~ bs/X~ei7T/4 as Ai -» oo where 6 is some positive constant, whereas gr(0) < 0, 5i(0) = 0. Thus A^ioo^g is either — |7r or | t t , depending on whether the pi (A*) is positive or negative, respectively. But by scaling, Ci(Xir) —> 0 as r —> 0 and Ci(Air) -> oo as t -> oo, with Ai ^ 0. Thus we have: <?;(A*) > 0 for r sufficiently small and <?i(A*) < 0 when r is sufficiently large. In particular this shows the existence of rh at which g(X*) = 0. Moreover, this shows that M = 0 for small enough 18 (1.33a) (1.33b) (1.33c) r and M = 2 for large enough r . Note that this does not show the uniqueness of t^ . That is, we cannot rule out the possibility ofgj(A*) changing sign more than once as r is changed. Nonetheless, by choosing the largest such Th, this shows the existence of for which A is purely imaginary, and such that there are two positive eigenvalues for all r > r^. To show the existence of r + , some algebra shows that C"(rA) < 0. It follows from convexity of / and the fact that C —> oo as rA —> oo and C(0) < 1, that for small enough r, the curves C and / do not intersect, and for large enough r, they intersect twice. Thus by continuity, there must be a value of r = T+ where these two curves intersect tangentially. Moreover, this value is unique because of convexity. For r > r + , these intersections are clearly positive. This concludes the proof of Theorem 1.1.9. • 0 50 100 150 200 t Figure 1.5: Plot of vm = i/(0) versus t for a one-spike solution with e = 0.01,1 = 1, D = 0.1, A = 6.59. The heavy curve corresponds to r = 8.6, the lighter to r = 8.8. Note that the proof of this theorem is constructive: it gives a method for computing provided that fi,'fr are available. To verify this theorem, we took a one-spike solution with parameter values I = 1,D = 0.1, A = 6.59 with e = 0.01. For this parameter set, we have computed numerically that = 8.7. We then plotted the evolution of the one-spike solution in time 19 for r = 8.6 and r = 8.8. The result is shown on Figure 1.5. For r = 8.6 we get decaying oscillations in spike height, and for r = 8.8 the oscillations start to increase in magnitude, eventually leading to the collapse of the spike. Thus the theory gives a correct prediction for the Hopf bifurcation value. Moreover, the increase of amplitude for r > suggests that the Hopf bifurcation is subcritical. Unfortunately the linear theory is insufficient to explain this. 1.1.4 Hopf bifurcation in the intermediate regime Next we consider the intermediate regime (1.3), 1 <S A < e-1'2. From (1.27) we obtain, for the small solution u~,v~: A2 First, note that if Ar = 0(1) then we have Cj ~ ^ m which case the N L E P problem (1.18) is stable by Lemma 1.1.3. On the other hand, if 9\ is large, (1.20) becomes: Cj ~ \ + ^VT\ coth(Wo/k). This suggests the following scaling for r : r = T0S2 tanh 2 {W0/k) so that Cj becomes: Cj 2 ' and is independent of j. The proof of Theorem 1.1.9 then goes through as before, and shows the existence of the Hopf bifurcation ro = ro^, as well as the existence of ro = ro+ at which the eigenvalue merges with the real positive axis. Numerically, we find: T0H = 1.748, Xih = 0.534; r 0 + = 8.567; A+ = 0.300. (1.34) We summarize. P ropos i t i on 1.1.11 Consider the k spike small solution in the intermediate regime 1 <S A <C e~1/2, with S> 0. Such solution is stable with respect to large eigenvalues whenever r <C 0(Ai). It undergoes a Hopf bifurcation at TH = A4D/9t&nh4(l90/k)T0H (1.35) 20 where TQ^ is given by (1.34). At that value of r, 2k large eigenvalues cross the imaginary axis at asymptotically the same time. Note that since the Hopf bifurcation for all the different nodes j occurs simultaneously, and since the corresponding k eigenvectors cmj given by (1-24) span all of Rk, the oscillating motion of the different spikes is independent of each-other. While the expression (1.35) is valid asymptotically as e —* 0, numerically we find that it provides a poor approximation for moderate values of e such as e — 0.05. A better agreement with numerics is obtained by finding the second term in the asymptotic expansion for r^. This calculation was performed in [74] with the following result: r h = ^ / 9 t a n h > o A ) r o , ( l - ^ j ' . 1.2 High-feed and intermediate regime We continue our study of Gray-Scott model (1.1), but this time we consider intermediate and high-feed regimes (1.3), (1.4). We find two types of instabilities that occur there, which are not present in the low-feed regime. The first type is an undercrowding or pulse-splitting instability. That is, when A is increased past a certain threshold, some of the spikes will be split into two spikes that start moving away from each other. This process may repeat a number of times, until sufficiently many spikes have been generated, at which time the system reaches an equilibrium. A n example of such instability is shown on Figure 1.2:a. The pulse-splitting occurs only in the high-feed regime, and corresponds to the dissapearence of the steady-state solution as A is increased past a certain threshold. In §1.2.1, we derive the minimum number of pulse splittings that will occur before the spikes settle to a new equilibria. The second type of instability is the slow oscillatory drift instability, as shown on Figure 1.2b. This instability occurs in both intermediate and high-feed regimes, whenr is of 0(\). The cause of such an instability is the existence of small eigenvalues of 0{e) which cross the imaginary axis into the positive half-plane as r is increased past a certain threhold. In §1.2.2 we derive 21 an expression for this threshold. For the rest of this section, by scaling x = x\/rD, we will assume without loss of generality that D = l. To obtain results for an arbitrary D , it suffices to replace e by ^= and I by =̂= in the results that follow. 1.2.1 Equ i l i b r i a in high-feed and intermediate regime We begin with the analysis of equilibria solutions of (1.1) with D = 1: 0 = £2vxx — v + Av2u 1 = V-xx — U + 1 — V2V v'(±l) =-0 = u'(±l) xe[-l,l] 0 = uxx - u + 1 - v2u • (1-36) We have the following result: P r o p o s i t i o n 1.2.1 Let B = tanh ( -[ ) A. (1.37) Suppose A e1!2 and B < 1.35. Then there exists a spike solution to (1.36) of the form v~-V0(r), u~^-U0(r), r = - . £ A £ where Vo(r), Uo(r) satisfy (1-4%) below with boundary conditions Vo(0)=0 = Uo(0), V0(oo) = 0, t^(oo) = B . (1.38) Conversely, if B > 1.35, then the k-spike solution of this form does not exist. Thus, for any given k, the /c-spike solution will dissapear if A is sufficiently increased. Con- versely, for any given A or domain size I, there will be a lc spike solution if k is large enough. For this reason, we refer to this instability as undercrowding or pulse-splitting instability. This 22 type of instability is in some sense the opposite of the overcrowding instability discussed in Theorem 1.1.7, where the equilibrium solution becomes unstable if k is too big. Before showing Proposition 1.2.1, we consider a numerical example. Consider the case I = 2, A = 3.5, D = 1, e = 0.03, r = 1. Letting Bk = A t a n h ^ , we obtain: B1 = 3.34, B2 = 2.66, B3 = 2.03, B4 = 1.61, B5 = 1.32. We see that B4 > 1.35 but B5 < 1.35. So for the equilibrium to exist, we must have k > 5. Indeed, starting with the initial conditions of one spike whose center is located slightly asymmetrically at XQ = 0.1, the system undergoes 4 additional spike splittings, and then settles to a 5-spike equilibrium, as shown on Figure 1.2a. To show Proposition 1.2.1, we start by constructing a single-spike solution on [—/,/]. Because of the Neumann boundary condition, we can then glue together k copies of such a solution to obtain a /c-spike solution on [—Ik, Ik]. By redefining / —> l/k we will then obtain a k spike solution on the interval [—M]- Near the core of the spike, we make the following rescaling: v=-V(r), u=^rU(r), r = - . (1.39) The steady state solution then satisfies: 0 = V" - V + V2U (1.40a) 0 = U" - £2U + Ae - V2U. (1.40b) Therefore we expand: v = -(V0(r) + AeV1(r) + ---), u = - (U0(r) + AeU^r) + • • •) '(1.41) £ j\. to obtain: 0 = K -V0 + V2U0 (1.42a) 0 = Ud- V02U0 (1.42b) 23 and 0 = V" - Vi + 2V0U0V1 + V02Ui • 0 = U'{ + 1 - 2VQUQVX - VQ2UX. Next, we expand the outer solution as: u = UQ(XX) + eux + ... to obtain: UQ — UQ = — 1 + 1> U from where uo(xi) ='1 — J G{x\,x'i)v2{x'iju{x'i)dx'i where G is the Green's function with Neumann's boundary conditions: Gxx - G = -5(x - x'), Gx(-l,x') = 0 = Gx(l,x'). It is easy to see that 1 f J!(a;)J 2(x') if x < x' (jr\X X ) —  1 \ j ; ( x ' ) J 2 ( x ' ) - J i (x ' ) / 2 (x ' ) | JX(X>)J2(X) if a;'<X where J\(x) = cosh (Z + x ) , J 2 ( r ) = cosh (Z — x ) . Thus we have: / 1 (0)J 2 (0) - J x (0)J 2 (0) = 2Zsinh(Z)cosh(0, G(0,0) = ^coth(Z). and so: u{er) = 1 - 1 G(er,er') [itft/n + (2V0E/oVi + V 0 2 ^ i ) ] ^ 1 f°° . U 0 (0) = 1 - - G ( 0 , 0 ) y_ V 0 2 t / 0 , 1 r 0 0 "o(0 ± ) = - I G x ( 0 ± , 0 ) y_ l / 0 2 C/ 0 , " o ^ ) = - 7 ^ ( ^ , 0 ) / V02U0 = - - G ( 0 , 0 ) / l / 0 2 t / 0 . 24 We also have the following matching condition: This yields: Next, we define: uo(er) + eui(er) ~ -^U0(r) + e 2(7i(r). (1.47) u0(0) = 0, (1.48a) U^(±oo) = Au'o^), (1.48b) f/f(oo) = < ( 0 + ) (1.48c) 1 f°° B = ^l V02UQ. (1.49) —oo Equations (1.46b) and (1.48a) then yields: B = 2GJm=tanh®A- (L50) and from (1.48c), (1.48a), (1.46d) we obtain: (7{'(oo) = u0'(0+) ~ u 0(0) - 1 ~ - 1 (1.51) Integrating (1.42b) (or alternatively, using (1.46c, 1.48b) and Gx(0+,0) = - ± ) we obtain: poo Oo(°°) = / VoUo = B. Jo Next, we argue numerically that the solution to the problem (1.42) with boundary conditions Vb(0) = U0(0) = 0, Vu(oo) = 0, U0(oo) = B, exists only when B < 1.35. Multiplying (1.42a) by VQ, integrating and using VQ(0) = 0 yields: 1 f°° 1 ttf (0) - ^Vo\0)U(0) = J -V0(rfU^r)dr. Assuming V0 > 0 and integrating (1.42b), we see U'Q > 0 so that that J 0 °° iVb(r) 3(7'(r)(ir > 0. Therefore letting 7 = Vo(0)E/0(0), (1.52) we obtain 0 < 7 < | . For each value of 7 in that range, we compute numerically the corre- sponding value of B. The resulting plot is shown in Figure 1.6. This graph has a fold point 25 6.2 OA 0.6 0.8 1 l l2 1.4 7 Figure 1.6: The graph of 7 = V(0)C/(0) vs. B = 7'(oo). The fold point occurs at 7 = 1.02, B = 1.35. The dashed curves represent asymptotic approximations, derived in Propositions 1.2.2 and 1.2.3. Subfigures show the actual solution for three different values of 7 as indicated. In subfigures, the thicker curve represents VQ and the thinner Uo(r) — (7o(0). Note that near the left endpoint, the solution looks like two well-separated bumps, whereas near the right endpoint the solution consists of a single bump: At 7 = 1, the solution is flat at the top. at 7 = 1.02, at which point B = 1.35. Beyond B = 1.35, no single-spike solution exists. This concludes the proof of Proposition 1.2.1 • In general, the problem (1.42, 1.38) must be solved numerically. However asymptotic expansion is available in the intermediate regime, when B is small. We first study the limiting behavior 7(0) 3> 1. We introduce a small parameter 8 by 6 = 1/(7(0), and expand the solution to (1.42) in terms of 5 as V = 5 (v0 + S2vi + •••), U = 5'lu = 8"1 (u0 + 52ux + •••) , with u(0) = 1. (1.53) Since J 0°° V2U dr = B from (1.49), the expansion (1.53) yields that B = 0(8). Hence, we write B = 5B0. Substituting (1.53) into (1.42), and collecting powers of 5, we obtain the leading-order 26 p r o b l e m v0 - v0 + VQU0 = 0, U'Q=0, ( I - 5 4 ) w i t h v'Q(0) = 0, i to(0) = 1, vo —> 0 as r —* c o , a n d u'Q —> 0 as r —» oo . T h e r e f o r e , w e o b t a i n uo{r) — 1 a n d vo(r) = t c ( r ) , w h e r e io sat isf ies (1.7) . F r o m c o l l e c t i n g t e r m s o f o r d e r 0(52), we see t h a t v\ a n d u\ sa t i s fy Lv\ = u i — v\ + 2wvi = — w2u\, 0 < r < o o ; D 1 ( 0 ) = 0 , «I —• 0 , as r —» o o , (1.55a) u x = w2 , 0 < r < c o ; u ' 1 (0) = 0 , t t i ( 0 ) = 0 , u\—> Bo, as r —> o o . (1 .55b) B y i n t e g r a t i n g (1 .55b) over 0 < r < oo, we get t h a t Bo = /0°° u? dr = 3 . N e x t , we m u l t i p l y (1.55a) b y w a n d in t eg ra t e over 0 < r < oo . T h e n , u s i n g Lw — 0 a n d w'(0) = v[(0) = 0, we get f°° / // f°° i / w Lvi d r = w (0)vi (0) = — / w2w u\dr . (1.56) Jo 7o I n t e g r a t i n g t h e l a s t t e r m i n (1.56) b y p a r t s , a n d n o t i n g t h a t u i ( 0 ) = 0, we get « ; " ( 0 ) u i ( 0 ) = - / i o 3 u i d r . . (1.57) 3 Jo T o c a l c u l a t e f i ( 0 ) , we m u s t d e t e r m i n e u^r). T o d o so, we s u b s t i t u t e w(r) = | s e c h 2 ( r / 2 ) a n d w2 — w — w d i r e c t l y i n t o (1 .55b) t o get // // 3 o / , . " / o \ ux — w — w = - sech {r/2) — w . (1.58) I n t e g r a t i n g t h i s e q u a t i o n t w i c e , a n d u s i n g ^^(0) = 0, w i ( 0 ) = 0, w e r e a d i l y . o b t a i n u i ( r ) = 6 l n c o s h s e c h ' ( i ) + T o d e t e r m i n e t>i(0), we i n t eg ra t e (1.58) once t o get 'r Ui = 3 t a n h / 3w i w = — w . w (1.59) (1.60) F i n a l l y , s u b s t i t u t i n g (1.60) i n t o (1 .57) , i n t e g r a t i n g t h e r e s u l t i n g e x p r e s s i o n b y p a r t s , a n d u s i n g t h e e x p l i c i t f o r m o f w, we o b t a i n 1 f00 / \ 1 "i(0) = sum I y~3w2w'" ™v)dr = (0) w 3(0) + 33 16 (1.61) 27 Since 7 = (To(O)Vo(O), we use (1.53) to get 7 = w(0) + S2vi(0). We summarize the result of this calculation in the following formal proposition: Proposition 1.2.2 Consider the intermediate regime B = 2>8, with 8 1, where B = A tanh(Z//c). Then the core problem (1-4-2) admits a solution with 7 = (7o(0)Vo(0) —> 3 / 2 _ as 5 —> 0. T/iis solution is given asymptotically by V0~S(w(r) + 82v1(r) + ---) , U0 ~ 5" 1 ( l + 82ux(r) + • • •) , (1.62a) where w(r) satisfies (1.7), t t i ( r ) is given in (1.59), and v\ satisfies: vl ~~ vl + 2wvi = —w2ui, Vi(0) =0, v\ —> 0 as r —•> 0 0 . (1.62b) Moreover, we have fi(0) = — f|, 7 ~ ^ - | * 2 + - - - . (1.62c) The asymptotic curve 5 ~ 35,7 ~ | — ||c5 2 is shown as the dashed parabola in Figure 1.6. As seen from that figure, it provides a good approximation to the actual solution of the core problem (1.42) in the limit 7 -+ 3/2". Next we construct a solution that corresponds to the regime 7 ^ 0 . First, note that V0"(0) = 0 at 7 = 1 and Vo"(0) > 0 when 7 < 1. This suggests that in the limit 7 —> 0, the solution has the shape of two bumps. To analyze this regime, we again label 5 = 1/Uo, with Uo = U(0), except that now we make a two-spike approximation for V with spikes located at r = r\ > 0 and r = —r\. As 5 —> 0, we will show that r\ ~ — ln<5 3> 1. Therefore, the separation between the spikes grows as 5 decreases. The analysis below to calculate r\ is similar in essence to the analytical construction of multi-bump solutions to the Gierer-Meinhardt model (cf. [26]) given in [12]. We look for a two-bump solution to (1.42) in the form V0 = 5(w1 + w2 + i ? + • • - ) , U0 = (5 _ 1tf = ( T 1 (1 + 82ui + • • •) with ui(0) = 0. (1.63) 28 Here we have labelled wi(r) = w(r—ri), W2(r) = w(r+ri), and we assume R <C 1. Substituting (1.63) into (1.42), we obtain that u\ and the residual R satisfy LR = R — R + 2(w\ + VJ2)R = —2wiu>2 — S2 [w\ + w2 + 2wiu>2) u\ , —oo < r < oo, (1.64a) Ui' = it;2 + w\ + 2w\Wi, — oo < r < oo , (1.64b) with u'-^O) = 0. Notice that wiw2 is small when the bumps are widely separated. In particular, when TI = 0(— In 5), the two terms on the right hand-side of (1.64a) are both of order 0(52). This is compatible with R being small, indeed it shows R = 0(S2). To determine ri we use a solvability condition. Assuming that the spikes are well-separated so that ri 3> 1, we multiply (1.64a) by u ^ , and then integrate by parts over —oo < r < oo to obtain the solvability condition 0 = J WiLRdr ~ — 2 J wiw 1W2 dr — S2 J w\wiUidr . (1.65) The dominant contribution to the first integral on the right hand-side of (1.65) arises from the region where r = r-_. In this region, we use w(r) ~ 6e~r to get w(r + 2r{) ~ 6e~r~2ri. In this way, we calculate Ii = 2 J W1W1W2 dr ~ 2 J w(r)w (r)w(r + 2r{) dr ~ 1 2 e _ 2 r i J e~rw(r)w (r) dr . (1.66) Integrating the last expression in (1.66) by parts, and then using w2 = w — w", we obtain h ~ 6 e " 2 n j e~r (w - to") dr = 6 e - 2 n lirn^ [e" r (w + w'^ = 72e-2ri . (1.67) To calculate the other integral on the right hand-side of (1.65) we integrate by parts once and use ui(0) = 0 to get f t 52 f 52 f I2 = S2 w\w'iUi dr = —— I wfu'i dr = -— [w(r)}3 u\(r + r{) dr . (1.68) Next, by integrating (1.64b) with 1x^(0) = 0, we get i t i ( r ) ~ f [w(s -ri)}2 ds. (1.69) Jo 29 We can then write u\ (r + r\) as u i ( r + n ) ~ / [w(s)}2ds+ [ [w(s)}2ds~3 + [ [w(s)}2 ds. (1.70) J-ri JO Jo Here we have used J ^ r i w2 dr ~ w2 dr = 3 for n >̂ 1. The integral on the right hand-side of (1.70) is odd. Therefore, upon substituting (1-70) into (1.68), we get an integral that is readily evaluated as I2 ~-S2 J [w(r)f dr = . (1.71) Finally, substituting (1.71) and (1-67) into (1.65), we obtain that r\ satisfies 7 2 e - 2 n „ ^ 1 . (1.72) 5 The product 7 = UQVQ is calculated as 7 ~ [u>i(0) + 102(0)] = 2w(ri). Since r\ 3> 1, we use w(r) ~ 6 e _ r to get 7 ~ 1 2 e _ r i , where n satisfies (1-72). Finally, B is determined by B — 5 JQ°° \W\ + w2,] ^ r ~ 65. This formal construction of a two-bump solution is summarized as follows: Propos i t i on 1.2.3 Let 5 = 1/U0(0) < 1,7 = U0(0)V0(0). Then the core problem (I.42) admits, a two-bump solution with 7 —> 0 + as 5 —> 0 + . This solution is given asymptotically by VQ~5[w(r-ri) + w(r + ri)] U0 ~ 1 ( l + S2 [Ul(r - n) + u i ( r + n)]) , (1.73a) where w(r) satisfies (1.7), and ui(r) is given explicitly in (1.59). The constants 7, r\, and B, are given for 5 <C 1 by B = 6S + ---, 7 = t/"oV0 ~ , r i ~ - l n < J + J l n l O . (1.73b) V5 2 T/iis implies the following local behavior for the B = B(y) curve: 7 = UQVQ ~ V2B/V5, as B —> 0. (1.73c) Numerically, the single-bump solution found in Proposition 1.2.2 joins with the double-bump solution of Proposition 1.2.3. Since B —> 0 at the endpoints 7 = 3/2, 0, this necessitates the 30 existence of the fold point corresponding to the maximum of the B = B(^) curve. However it is an open question to show this rigorously. Next, we argue numerically that the existence of a connection between a single and double- bump solution graph causes the pulse to split into two when B is increased beoynd the the fold point at B = 1.35. We first show the existence of the zero eigenvalue at the fold point. We introduce a perturbation around the equilibrium state: v{x) = I (V(r) + eAt$(r)) , u(x) = (l/(r) + eXTN(r)j where r = - to obtain: A$ = - $ + V2N + 2VU& (1.74) T£2XN = N" — s2N — V2N — 2VU<&. At the fold point 70 = 1.02, B = 1.35, we have ^ = 0. Using this identity and differentiating (1.42) with respect to 7, we obtain a leading-order solution to (1.74) given by A = 0, N = p . 070 «7o Next, we compute the function ^ numerically, usingthe approximation ^ ~ 200(Vo|70+o.oi — VQ\ 70-0.01 )• As can be seen from the resulting graph on Figure 1.7, this eigenfunction has a dimple-like shape. As a consequence, when B is increased just above 1.35, the equilibrium solution dissapears, but the shadow of the dimple eigenfunction will control the resulting dy- namics. Its shape has the effect of splitting the spike into two. Numerically, immediately after splitting the two spikes start to move away from each-other. As they move far enough apart, the equilibrium state may again dissapear and the whole process can repeat itself. 1.2.2 Slow drif t and osci l latory dri f t instabi l i t ies So far, we have studied instabilities which occur on a fast time scale, due to an unstable eigen- value of O(l)• Next, we study instabilities with respect to small eigenvalues. Such eigenvalues 31 Figure 1.7: The graph of the dimple eigenfunction $ = ^ at the fold point 70 = 1.02. exist because of translation invariance of the Gray-Scott model. The corresponding perturba- tion is the derivative of the profile to the leading order. The main result of this section is the following. T h e o r e m 1.2.4 Let sA<^ 1, A^> e 1 / 2 and e C 1. Then the small eigenvalues X associated with drift instabilities of the k-spike equilibrium solution of Proposition 3.1 are of order O (eA), and satisfy the k transcendental equations, A ~ as A j = 1,..., k, (1.75) where ex = VI + XT and a is some positive constant that is defined in (1.102) below. This constant depends only on the bifurcation parameter 7 associated with the core solution, and is listed in Table (1.80). The constant a is found to be positive numerically. In the inner region near the m^1 spike, the perturbation of the v-component of the equilibrium solution that corresponds to j-th small eigenvalue (1.75) has the following form: v(x, t) ~ i (Vo {^j^j + PcmjVl ( ^ Y ^ ) eAt) , |s - xm\ « 1 (1.76) 32 where Q <C 1 is some small constant, VQ is defined in Proposition 1.2.1, and 7T7 cmj = sin ( — (m — 1/2) In the intermediate regime A ~ 3coth(l/k)5 with the constant a is given asymptotically as a ~ 25 ~ -Atanh(Z//c) 7 B f*\Vr a 1.48 0.30 -10.01 0.20 1.46 0.40 -7.51 0.27 1.44 0.50 -6.00 0.33 1.42 0.60 -4.98 0.40 1.39 0.70 -4.24 0.47 1.36 0.80 -3 .67 . 0.56 1.32 0.90 -3.20 0.63 1.28 1.00 -2.79 0.72 1.23 1.10 -2.41 0.83 1.17 1.20 -2.01 1.00 1.14 1.25 -1.77 1.13 1.10 1.30 -1.48 1.35 1.02 1.347 -0.74 2.70 We begin by linearizing around the equilibrium ve,ue : v = ve + eXt(f>, u = ue + eXtr]. Substituting (1.81) into (1.1) with D = 1 we obtain the eigenvalue problem £2<l>xx -<t> + 2Aueve(f> + Arjvl = Xcj), - 1 < x < 1, (1.77) (1.78) (1.79) (1.80) Vxx-V - VVe - 2ueVe(j) = TXr] . -I <x <l . (1.81) (1.82a) (L82b) We begin by examining (1.82) in the inner region where r = e^1(x - Xj). Since we use the scaling (1.39) for the equilibrium solution, we also rescale the eigenfunctions as cj)(x) = - $ ( r ; e ) , n(x) = eN (r; e) . (1.83) 33 to obtain (1-74). Next, we expand <I>, N and A in eA: <E> = $o + A e $ i , N = N0 + AsNi, A = AE\Q. Note that V2 = V2 + 2eAV0V1, UV = UQVQ + eA(V0Ux + UQV{) where Um, Vm are defined in (1.42) and (1.43). Thus we obtain: 0 = $'0' - $ 0 + Vo2iVo + 2V0U0$0, (1.84a) 0 = < - V2N0 - 2V0U0$0. (1.84b) and: A 0 $ 0 = $'/ - + V"02JV"i + 2V0VlN0 + 2Vot/o*i + 2(V 0 t / i + C/ 0Vi)So (l-85a) 0 = iV{' - .[V0 2JVi + 2V 0ViiVo + 2V0U0i>1 + 2 ( V 0 ^ i + CW )$o] . . . (1.85b) Differentiating (1.42), we see that (1.84) admits a solution: $0 = 0,%', N0 = CjU^ (1.86) where Cj will be determined below through asymptotic matching. Substituting (1.86) into (1.85), we then express the result in matrix form as / A 0 0 \ / Vr \ ( -2{UVX + UXV) -2VVX \ ( Vr = + <* I ' ( L 8 7 a ) \ 0 0 J \ Ur J \ 2(UVX + UXV) 2VVX J \Ur where * = ( $ i , Nxy, and the operator C is defined by I * i r . \ ( $ i \ ( -1 + 2UV V2 , / ^ \+E E=[ | . (1.87b) \ 7V L R R J \NX J V ~ 2 U V - V < 1 To determine the solvability condition for (1.87), we let ^ denote the solution of the homoge- neous adjoint equation £ t * t = ( \ + E t \ 1 \=0, (1-88) 34 where t denotes transpose. We look for an odd solution to (1.88) where ^ \ —* 0 and —> 0 as |r| —> oo. To determine the solvability condition for (1.87a), we multiply (1.87a) by and integrate by parts to get J * t4£* dr = (tf t** r - 'tfj) 1^ = CjXQh + Cjh , (1.89a) where I\ and I2 are defined by h = J _ ^ [2(UV)rVx + (V2)^} dr, h = J *\vr dr . (1.89b) Since \£ and \&t are odd functions and ^/J —» 0 as r —> oo, (1.89a) can be reduced to CjXoh = -Cjh + * 2 ( ° ° ) (Nir(+oo) + Nlr(-oo)) . (1.89c) CW = (1.90) The next step in the analysis is to calculate I\ explicitly. To do so, we introduce W by W = (Vir,Uiry. Upon differentiating the system (1-43) for V\ and U\ with respect to r, it follows that ' -i<yv\vx - ( y 2 ) r r j ! 2(UV)rVx + ( V 2 ) ^ Therefore, J i in (1.89b) can be written in terms of W. Integrating the resulting expression by parts, we get h = j ¥lCW dr = (V'W,. - 1^ + j WlC]¥ dr . (1.91) Using £tv[/t = 0, with -+ 0 and <b\r -> 0 as |r| —> oo, (1.91) reduces to h = 2*^(oo)(7i r r(oo) = -2#J(oo). (1.92) Here we have used (7 l r r.(co) = - 1 as seen from (1.51). Finally, substituting (1.92) into (1.89c), we obtain a compact expression for Ao "1 CjAo J V\Vrdr = 2*+(oo) (7V l r(+oo) + iV l 7 .(-oo)) + Cj j = l,...,k. (1.93) 35 This completes the analysis of the j ^ 1 inner region. Next, we match this inner solution constructed above to the outer solution for (1.82). This will determine ATi r(±oo) and the constants Cj, for j = 1,.. . , k. In the outer region, we have: Vxx ~ (1 + i~\)r) = -nvl + 2ueve(f). (1.94) Away from 0(e) regions centered at the spike equilibrium locations, ve is exponentially small. Therefore we have: Vxx ~ (1 + T\)r] ~ 0, Xj (1.95) Matching the inner and outer solution, we have: e (cjUr(±oo) + eANx) ~ r)(xf) + err]x{xf) + Using Ur(±oo) = B we therefore obtain: il(xf) = ±ECjB, 7Vi r(±oo) = -^Vx(xf) (1.96) (1.97a) (1.97b) We now solve (1.95) on each subinterval and we use the condition (1.97a) for r](x- ) and the boundary condition r / x ( ± l ) = 0. This yields, cBcL coshf^(i+^)l. Lcosh[6>x(l+xij] ' r/(x) = { eBCjsinlZf-Xj+l)i - eBcJ+1 .sirfS[^(-"aj)1>) v ' j J smh[t)(Xj— Xj+i)\ J i 1 smh[0\{xj+i — Xj)\ R cosh[g A(l-x)] £ i 3 Cfccosh [ e A(l-x f c)] ' Here 6\ is defined by 6X = V l + rA. Combining (1.98) and (1.96) we obtain, for j = 1: (JVi r(+oo) + JVi r (-oo)) = B6x 2 2A — l<X<Xl, Xj < x < Xj+i, j = 1,... ,k - 1, X). < X < I . (1.98) (1.99) -ci coth —-— + tanh V w > — C2 csch 210, (1.100a) 36 For j = 2 , . . . , k — 1, we get (JVl7-(+oo) + 7V l r (-co)) = B9X 2 2A Similarly for j = k, we get (iV l r(+oo) + iVi r ( -oo) ) = B0x 2 2A - C j _ i csch [ —— 1 - 2cj coth I —̂ — J - Cj+\ csch I — ^ (1.100b) , ,'2Zt?A\ / / 2 Z 0 A \ , (19X -Ck-i csch —— — cfc coth —— + tanh ' (1.100c) Substituting (1.100) into (1.93), and using the relation B/A = tanh (Z0o/fc) from (1.37), we obtain the following matrix problem for c = ( c i , . . . , c^)* and Ao: 'I Anc = a Here a is defined by and B is the tridiagonal matrix a = d / 0 / e / 0 / e 0 0 0 0 0 0 0 0 0 \ with matrix, entries ci, e, and / defined by d = coth [ + tanh (jjf) , e = 2 coth 0 0 0 0 0 0 0 0 0 e / 0 / e f 0 f d ) 216X (1.101) (1.102) (1.103a) / = csch 219X (1.103b) k r J - ™ \ k The spectrum of B can be calculated explicitly as was done in Appendix D of [40]. This leads to the following result: 37 L e m m a 1.2.5 The eigenvalues Q, ordered as 0 < ("l < ••• < Ck, of B and the associated eigenvectors Cj of B are 4 = 1,-1,1, . . . , ( - l ) f c + 1 ; c m J = sin - f ( m - l / 2 ) , j = l , . . . , k - l . (1.104b) Here c* denotes transpose and c* = ( c i j , . . . , Ckj). Substituting (1.104) into (1.101) leads to (1.75). As shown in Table (1.80), the constant a is always positive for the primary branch 7 > 1.02. For general B, this constant is determined numerically. However in the intermediate regime B = 3<5 < C 1 this constant can be determined analytically. In this limit, we recall from Proposition 1.2.2 that Vb ~ 5w, and U0 ~ 5~l. Here w is the spike profile given in (1.7). Therefore, for 5 < C 1, the homogeneous adjoint problem (1.88) reduces to The solution to this limiting system is odd and is given, up to a normalization constant, by (1.104a) (1.105) (1.106) Therefore, for 5 <§; 1, we have (1.107) To calculate the integral in (1.107) we use w(r) = | sech2 (r/2) to obtain a = 25. This concludes the proof of Theorem 1.2.4. • Note that if r < ^ then 6\ ~ 1 in (1.75), as A = 0(e). In this case it is easy to show that (1.75) reduces to A ~ — eaA sech2 (l/k) l - i ( l + COS(7TJ/fc)) , j = l...k. (1.108) We thus have the following corollary: 38 C o r o l l a r y 1.2.6 Suppose that r <S ^. Then the small eigenvalues in the high-feed or inter- mediate regime, as given by Theorem 1.2.4, are all stable, and are given explicitly by (1.108), where a > 0 is defined in Theorem 1.2.4- We now show that when r = O(^), the small eigenvalues can be destabilized via a Hopf bifurcation. To analyze (1.75)'it is convenient to introduce the new variables Td, u>, and £, defined by \ = £aAcj, T=(j^j)T<l' i = rdu. (1.109) Substituting (1.109) into (1.75), we obtain that £ satisfies: | - = <3(0 = / ^ ( O T ( c o g h ( / 3 z ) + T ) _ l , j = 1.. . fe, where (1.110a) z = y/l'+ £, B = 2l/k, 7 = cos ( ^ J (1.110b) Elementary calculus yields the following facts about G, j — 1 . . . k: G ( 0 ) < 0 , G ' ( O > 0 , G'j(t)<0. (1.111) It follows that (1.110) has a negative double root for some = Td- sufficiently small and positive, and has a positive double root for some = sufficiently large and positive. For Td € (T ( 2_ ) T ( J+) ) the two roots are therefore complex conjugate; and by continuity, they must cross the imaginary axis at some Td = T^H- This shows the existence of the Hopf bifurcation. Moreover, the following theorem shows uniqueness of Tdh for each j: T h e o r e m 1.2.7 Suppose eaA -C 1 and A 3> 0(e1^2), where a is given in Theorem 1.2.4- Then for each j = 1.. . k, there exists a unique positive r = Th = -^jjdh where Tdh — 0(1) such that the corresponding small eigenvalue A given by Theorem 1.2.4 is purely imaginary. Moreover, Re A < 0 if T < Th and Re A > 0 if T > Th- Proof. We set £ = in (1.110). Taking real and imaginary parts of (1.110a) we obtain: R e ( o ( * ) ) = o , ^=dlm (L112> 39 Some algebra shows that — Re(G(^ ) )>CJ , Im(G(i&)) > 0. (1.113) In addition we have Re(G(iO)) < 0, Im(G(iO)) = 0. (1.114) It follows that the solution to (1.112) is unique with the corresponding r& > 0. This proves the uniqueness of TVJ. The fact that Re A > 0 if and only if r > T>J follows from uniqueness of T& and from the continuity argument discussed before the statement of Theorem 1.2.7 ' [-] If Td is increased just to the right of r ^ - , the k spike solution is destabilized by a perturbation of the form (1.76). In the case of k = 1, this corresponds to slow oscillations of the spike around the equilibria location. In the case k = 2, j = 2, the corresponding eigenvector is c2 = (—1,1) and the shape of the perturbation causes a slow asynchronous oscillation of the two spikes around their equilibria. A n example of such behaviour is shown in Figure 1.2b. B y contrast, if k = 2,j = 1, then the oscillations around equilibria are synchronous. However we were unable to observe this type of oscillation numerically. Indeed, numerical computations suggest that the the asynchronous oscillation corresponding to the node j = k is always triggered before any other nodes. We state this as a conjecture: Con jec tu re 1.2.8 Let r^j be the Hopf bifurcation values as found in Theorem 1.2.7. Then the following monotonicity property holds: It follows that as r is increased, the asynchronous drift corresponding to the node j = k becomes unstable first. ' • Thus we do not expect to observe synchronous drift in the absence of asynchronous drift. We have the following evidence for this conjecture. Write G from (1.110a) as Thl> T h 2 > •••> Thk- G = a(b + 7) — 1 where a = z tanh (B/2) sinh (3z) b = cosh (3z). 40 Then £i and r^h satisfy: R e G = ar(br + 7) - a^i - 1 = 0, (1.115a) I m G = 0^7 + arbi + 6 ra; = (1.115b) Solving for 7 in (1.115a) and substituting into (1.115b), we obtain: 1 ai + bAal2 (1.116) As will be shown in the proof of Lemma 1.2.9 below, & 6 [0, £*), where £* is the first root of ar(i£i). Now in Lemma 1.2.9 below we will show that £i = Ci/ij i s a n increasing function of j. Thus we would have the proof of the conjecture if we can show that the right hand side of (1.116) is an increasing function of £j, at least for £j 6 [0, £*). Using a computer, we have verified this condition for various values of 0. It remains to show: L e m m a 1.2.9 The following monotonicity property holds: Here, £ihj is the unique purely imaginary solution £ = i^ihj of (1.110). Proof. Let ^ be the unique root of R e G ( i £ i ) = 0 as found in Theorem 1.2.7. We need to show that is a decreasing function of 7. The proof is illustrated on Figure 1.8, and is explained below. From (1.115a), it follows that the curves ReG(i£j) for different values of 7 all intersect at the same point whenever aT = 0, and do not intersect at all otherwise. Label the first such point by £*. But then ReG(z£*) > 0 since ReG( i£ i ) | 7 =i is always positive due to (1.113) and because of the fact that R e G ( i O ) | 7 = i = 0. Thus € (0,£*) for all 7 e (-1,1) . But a r(i&) > 0 on this interval since ar > 0 when £{ = 0. It follows that for € [0,£*), R e G is an increasing function of 7. This proves that its root ^ is a decreasing function of 7. • Finally, consider TVJ = for which the eigenvalue A given by Theorem 1.2.4 merges with the positive real axis. Since G(z) is increasing in 7, TVJ+ is a decreasing function of 7. Thus 41 Figure 1.8: Plot of ReG(z£;) for three values of 7 as indicated, and with 8 = 2. A l l three curves intersect at a the same point £* with Re G(i£f) > 0. corresponding to 7 = — 1 i.e. j = k, is the first value of r for which there exist purely positive small eigenvalues. The following table lists the value of r ^ , computed numerically, for various values of 7 and B, as well as the value for k = j. 42 p ' Tdh 7 = - l Tdh 7 = cos(vr2/3) Tdh 7 = 0 Tdh 7 = cos(7r/3) Td+ . 7 = - l 0.1 663 735 828 957 1688 0.2 167 185 208 240 423 0.3 74.7 82.78 93.19 107.6 189 0.4 42.5 47,12 53.00 61.14 107 0.5 27.66 30.62 34.40 39.64 69.20 0.6 19.58 21.65 24.30 27.95 48.59 0.7 14.71 16.25 18.21 20.91 36.16 0.8- 11.55 12.75 14.26 16.34 28.09 0.9 9.39 10.35 11.56 13.21 22.56 1.0 7.84 8.631 9.623 10.97 18.61 1.5 4.21 4.595 5.065 5.691 9.240 2.0 2.990 3.221 3.502 3.868 5.965 2.5 2.461 2.619 2.808 3.047 4.452 3.0 2.207 2.322 2.457 2.622 3.633 4.0 2.021 2.085 2.156 2.240 2.826 5.0 1.9831 2.018 2.056 2.097 2.461 6.0 1.9835 2.001 2.020 2.040 2.271 10 1.9989 1.9996 2.0003 2.0011 2.037 (1.117) This table provides a more direct numerical verification to Conjecture 1.2.8. Note also that r̂ , r + —> 2 as I —> co, for all values of 7. This is easily seen as in this case, we have G(0) '~ 0 and G'(0) ~ \ . Recall from Proposition 1.1.11 that profile instabilities in the intermediate regime may occur when T 1. Whether drift or profile instability occurs first depends on the scaling of A as we now show. For the profile instability in the intermediate regime, we have from Proposition 1.1.11: A4 m = ^2 tanh 4(Z//c)r 0/ l. On the other hand, we derived that the drift instabilities in the high regime regime occur when r is increased beyond 1 Using (1.79), in the intermediate regime this reduces to: 3 Ths = T o coth(l/k)rdh 43 Equating r^ s = T^I leads to the following threshold. P r o p o s i t i o n 1.2.10 Let A* = WE£l/6  c o t h 5 / 6^ f c)i / 6 ror =-l-404sV6 cot^(l/k)r^ where Tdh is defined in Theorem 1.2.7 and is listed in 1.117. If A < Ac then oscillatory profile instability occurs before the drift instability. If A > Ac then the drift instability occurs before the oscillatory profile instability. For the infinite line case I = oo, we have Ac = 1.578£ 1 / 6 . Full numerical simulations indicate that the Hopf bifurcation corresponding to oscillatory profile instability is subcritical, and as such, leads to the eventual collapse of the spikes. Thus in order to observe drift instability, we must take A > Ac- To verify the correctness of our theory, we consider several numerical examples. To perform numerical simulations in ID, we have discretized the problem in space, reducing it to n ODE's that are to be solved in time. We then used a code of [34] that implements an explicit Runga- Kut ta code of order 4-5 with stepsize control. E x a m p l e 1. We consider a one-spike solution with k = 1, A = 1.3, I = 2, e = 0.03, D = 1. For our initial condition, we took a spike slightly off-center, centered at x = 0.1. From (1.50) we have B = 1.25 and from Table 1.80 we find a = 1.13. Prom (1.110b) we have 0 = 4. From Table 1.117 we find that (rdh)j=1 = 2.02, rd+ = 2.8. From (1.109) we have r = 22.7r^ so that that the theoretical prediction for the oscillatory drift and the merging bifurcation is: Th ~ 46, r + ~ 64 44 Using direct numerical simulation of (1.1) with these parameter values, we observe that the spike is stationary for r < 55, whereas the slow oscillatory drift is observed for 58 < r < 73, settling into a periodic motion. For r > 73, the spike the drift instability causes an eventual extinction of the spike. See Figure 1.9. A n interesting behaviour is observed when r = 72. In this case, the amplitude of the drift is rather large. When the spike center approaches the edge of the domain, it triggers an oscillatory profile instability, so that fast oscillations in spike height are observed at that point. While the analysis of the oscillatory profile instability given in §1.1.3 was done for the spike at equilibrium position - and so is insufficient to explain the observed oscillations - it is possible to extend this analysis for spikes at arbitrary positions, as was done in [39] for the Gierer-Meinhardt model. 2 5 h I i / 57 • 66 72 • - • 74 II,./ \ 1 1 i l l i l 1 I: i 1 r - i i r i i l . I 1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2 r -2' 1 1 1 1 I i I I I i 0 200 400 600 800 1000 1200 1400 1600 1800 2000 t Figure 1.9: Example 1. Plots of the height and center of a single spike vs. time for four different values of r. Top figure: the height v(x0) plotted vs. time, where x0 is the center of the spike. Bottom figure: x0 vs. time. Here, k = 1, A = 1.3, 1 = 2, e = 0.03, D = 1 and r is taken to be 57, 66, 72, 74, as indicated in the legend. 45 Xj and x2 4 r 0 500 1000 1500 2000 2500 - 3000 3500 t x2+x ( and x 2~x ( 3r J L 0 500 1000 1500 2000 2500 3000 3500 f Figure 1.10: Example 2. Here, k = 2, I = 4, r = 70 and the other parameters are as in Example 1. Example 2. Next, we consider a two-spike solution with k = 2, / = 4 and other parameters as in Example 1. The equilibrium state is then the same as in Example 1, but copied twice. Thus we obtain, as in Example 1, (rdh)J=2 = 2.02, {rdh)j=1 = 2.16, rd+ = 2.8 Here, j = 2 corresponds to the asynchronous mode and j = 1 corresponds to the synchronous mode. Therefore we obtain: 7~h,asynchronous — 46, T f l < S y n c } l r o n o u s ~ 49, T_j_ ~ 64 In Figure 1.10 we plot the solution for r = 70. The top subfigure shows the positions of the centers x\ and x2 as a function of time. Initial configuration was taken to be two spikes located at 2.1 and -1.9. In the bottom subfigure we plot the sum and the difference in the position 46 of the spikes, respectively. A changing sum or difference indicates instability with respect to synchronous or synchronous mode, respectively. Since Th^asynchronous and Th<synchronous are within 3 units of each other, and since r = 70 was found about 13 units away from Th = 57 of example 1, we expect that both synchronous and asynchronous modes are unstable with r = 70. Indeed, for t < 1300, this appears to be the case as the amplitude of both x\ — x 2 and x\ + x 2 increases over time. However at a later time, the synchronous mode appears to die out, and asynchronous mode dominates. This appears to contradict the theory. A resolution of this paradox is that the oscillations happen far from the asynchronous bifurcation point and therefore linear theory is only applicable for relatively small values of t/e. 1.3 Discuss ion Most of the results presented in this chapter have been previously reported in [74] and [44]. The methods used in §1.1 were first derived in [40] and [77] to analyse instabilities in the Gierer- Meinhardt model. In addition to the large eigenvalue analysis of the low-feed regime of §1.1, it is also possible to carry out the small eigenvalue analysis as was done in [44] for the GS model, or [40] for the G M model. Unlike the intermediate and high-feed regime, the small eigenvalues can be unstable in the low-feed regime even with r = 0, when A is near the saddle-node point Ake and k > 1. Moreover, they become unstable through a saddle-node bifurcation, not through a Hopf bifurcation as in high or intermediate-feed regime. The corresponding small eigenvalues are always real, so one does not observe the oscillatory drift in such a case. In the low-feed regime, there also exists asymmetric equilibria, whereby the /v-spike solution at equilibrium may have spikes of precisely two different heights. As was shown in [74], asymmetric solutions bifurcate from the symmetric solution through a saddle-node bifurcation at precisely the same point where the small eigenvalues change their sign. These bifurcations are illustrated in Figure 1.11 Finally, the following equivalence principle between the one-dimensional Gierer-Meinhardt model (2) and the low-feed regime of the Gray-Scott model (1.1) is shown in [74]. 47 12.0 10.0 8.0 6.0 4.0 2.0 0.0 5.0 7.5 10.0 12.5 15.0 17.5 20.0 A Figure 1.11: Bifurcation diagram of symmetric and asymmetric spike patterns in the low-feed regime, for k — 1 , 2 , 3 , 4 . Solid curves represent symmetric solution branch and dotted curves represent asymmetric solutions bifurcating from the symmetric branch at the points •. The saddle-node values Ake, represented by *, increase with k. The part of the solid curve below (above) * is unstable (stable) with respect to large eigenvalues. The part of the solid curve below (above) • is unstable (stable) with respect to small eigenvalues. Proposition 1.3.1 Consider a k-spike equilibria solution to the Gray Scott model in the low- feed regime, as derived in Proposition 1.1.1. The eigenvalues of the linearized problem of the Gray Scott model are exactly the same as the eigenvalues of the k spike solution to the Gierer- Meinhardt model (2) with fl = [—1,1] and with the exponent set (p,q,m,s) = (2, s, 2, s), where s is defined in (1.18e). However the intermediate and high-feed regime of the Gray Scott model exhibit certain phe- nomena that have not been observed in the Gierer Meinhardt model. There have been several related works on Gray Scott model before [74] and [44]. Of particular interest to us is the work of Doelman and collaborators ([16]-[21], [55]) and the work of Muratov and Osipov [57]-[60]. The main difference between their work and [44], [74] is that they only consider the case of a single spike k = 1 on an entire space I = oo, whereas we also consider a bounded domain. Below we review and compare their results with ours. 48 We start with low-feed and intermediate regimes. The k spike symmetric equilibria corresponds to periodic solutions on the entire space. Existence of such periodic solutions has been rigorously proven in the low-feed and intermediate feed regime in [55] using dynamical system techniques. However their stability analysis is restricted to a single spike on the entire domain. In particular, they do not have the overcrowding instability threholds lk derived in Theorem 1.1.7. This instability is specific to having multiple spikes on a bounded domain in a low-feed regime, whose stability neither Doelman et al. nor Muratov and Osipov consider. In [18] and [19] Doelman et al also study the oscillatory profile instabilities for a single spike on the entire space, in the intermediate regime. They use the hyper geometric functions in their analysis of the corresponding eigenvalue problem. They also show the existence of the Hopf bifurcation which initiates the oscillatory profile instability, as we have done in Proposition 1.1.11. However their proof relies on the properties of hypergeometric functions. Our approach does not require their use. In addition we also consider multiple spikes, which they do not. The possibility of asynchronous profile instability (for example in the case of two spikes) is is specific to multiple spikes and does not occur with a single spike. Next we compare the results in the high-feed regime. The pulse-splitting has been numerically observed in [15, 17, 66]. However no analysis of it is given there. The first analysis appears in [57] where they also derive the dimple eigenfunction of Figure 1.7 at the fold point B = 1.35. In [57] they also observe numerically the connection between one and two-bump solution which we beleive is the underlying cause of pulse-splitting. However the Proposition 1.2.1, which predicts the minimum number of pulse-splitting events is new, as [57] only considers a single spike on an entire space. Finally, the drift instability in the intermediate regime has been analysed in [60], for a single spike on the entire space. In [16], Doelman, Eckhaus and Kaper also derive the equation of motion of the center of the spike in the intermediate regime, for the case of two spikes on the entire space moving away from each other. However in both of these, the drift is not oscillatory, because in the limit I —> oo, the Hopf bifurcation is replaced by a saddle-node bifurcation as Tdh ~ 2 ~ T + (see discussion after Table (1.117)). The oscillatory drift is specific to having 49 a finite domain. In addition, we also analyse the small eigenvalues for the high-feed regime, giving a hybrid analytic-numerical result of Theorem 1.2.4. Several open problems remain. In [77], the existence of the Hopf bifurcation with respect to oscillatory profile instabilities was proven. Numerical evidence indicates that it is unique, but we cannot show this at the moment. Another open problem is to prove Conjecture 1.2.8, which states that asynchronous drift instabilities always occur before the synchronous ones. Indeed, numerical simulations suggest a stronger result is true: the the end state is always an asynchronous oscillation, even with the synchronous mode unstable. A central open problem in the high-feed regime is to show the connection between one and two bump solutions, which would prove the dissapearence of the equilibria state, leading to pulse splitting. This connection seems to be a generic phenomenon. It is also present in the Gierer-Meinhardt model (2) when the ratio of diffusivities D/e2 is of 0(1) [69] as well as in the Gray-Scott model when 0(D) = 0(e 2 ) [63]. 50 Chapter 2 Stripe and ring-like solutions of the Gray-Scott model in two dimensions In this chapter we continue our study of the Gray Scott model (7). As in §1, we assume without loss of generality that D = 1. vt = e2Av — v + Av2u , x E ft ' rut = Au — u + 1 — v2u (2.1) dnv =• 0 = dnu, x e dfi. We consider stripe and ring-like solutions, where the domain is either a rectangle or a disk, respectively. Such solutions are one-dimensional in nature, and many of the techniques from Chapter 1 are then applicable with some modifications: We consider three types of instabilities of such solutions. First, a breakup instability, which results in a breakup of a ring or a stripe into m spots. Second, a zigzag instability, which causes the formation of a zigzag pattern. Finally, a splitting instability, causing a stripe or a ring to split into two. These instabilities are illustrated in Figure 2.1. We show that breakup and zigzag instabilities always exist for either the low, intermediate or high-feed regimes, whereas the splitting instability occurs only in the high-feed regime. The method of analysis of the zigzag instability is similar to the analysis of the slow oscillatory drift instability discussed in Chapter 1. Alternatively, the breakup instabilities on the other hand, are similar to the overcrowding instability for the low-feed regime, which was also discussed in Chapter 1. 51 - 2 - 1 0 1 2 - 2 - 1 0 1 2 Figure 2.1: Three different instabilities of a stripe: a single stripe splits into two at t ~ 0; a zigzag instability is visible at t ~ 70; followed by a breakup instability at t ~ 80. The end steady state consists of two spots and four half-spots. The domain size was set to [—2, 2] x [0,1], with £ = 0.07, A = 2 , r = 1. The results for the splitting instability for a stripe are equivalent to the corresponding results for a splitting instability of a one-dimensional spike. A similar technique also yields a corresponding result for ring splitting. In contrast to stripe splitting, the ring radius enters into the formula which involves Bessel functions. We also perform numerical computations in the regime where the ratio of the diffusivities is of O( l ) . We find the existence of labyrinth-like patterns and space-filling curves, such as shown in Figure 2.2. This chapter is organized as follows. In §2.1 we study the low-feed regime A = 0{ell2). In §2.1.1 we derive the equations of motion for a ring radius in the low-feed regime, using s a Melnikov-type calculation. We then prove the existence of a ring radius ro for which the ring is at equilibrium. In §2.1.2 we study the breakup instability of ring and stripe solutions in the low-feed regime. Stripe solutions in the intermediate and high-feed regimes are considered in §2.2. We characterize both zigzag and breakup instability bands for stripes in §2.2.1 and 52 t= 424 t=689 t= 917 -2 t= 1947 t= 4098 t= 17984 Figure 2.2: Development of a labyrinth-type pattern in the Gray-Scott model. Here, we take e = 0.5, n = [0,40]2, A = 2, r = 1. §2.2.2, respectively. For the intermediate regime explicit thresholds are derived whereas in the high-feed regime we present hybrid numeric-analytic results. In §2.3 we derive the analogous results for a ring in the high-feed and the intermediate regime. In the high-feed regime, we also derive a threshold on A above which the equilibria solution ceases to exist and ring splitting occurs. Some numerical simulations are given in §2.4. We conclude with a brief discussion section §2.5. 2 . 1 R i n g and str ipe solutions in the low-feed regime In this section we study the equilibria and stability of stripes and rings for the two-dimensional Gray Scott model in the low-feed regime A = 0(e1/2). Using the same scaling as in (1.5) and setting as before D = 1, our starting point is the following system: vt = £2Av — v + Av2u , x £ Tut = Au — u + 1 - \v2u (2.2) 'dni/ = 0 = dnii, x € dQ 53 where A = e ll2A. As usual, we proceed by first describing the equilibrium state and then looking at its perturbations. 2.1.1 R i n g equi l ibr ia solutions We first consider the equilibria solutions. Since the stripe equilibria is simply a trivial extension of a one-dimensional spike solution, in this section we concentrate on the less trivial case of a ring solution. We assume that the domain is a ball of radius R in RN, N>2: Q = BR = {x = (xi,X2,-•• ,XN) : | x | < i ? } . We will also allow the case of entire space R = oo. A ring solution is a solution that concentrates on a ring {\x\ = ro} of radius ro, as £ —> 0. In this section we derive an O D E which describes the slow evolution of ro in time. We then show that such O D E always admits an equilibrium in the case R < oo. When R = oo, the equilibrium may or may not exist, depending on the value of the parameter A. To derive the O D E for the ring radius r 0 , we start with the following anzatz for the solution: 1 (w(y)+sVl(y) + ...), y = ^ ^ , r=\x\, r 0 = r0(e2t), (2.3) AU0(0) u = U0(y)+eU1(y) + .... (2.4) The leading order equation for u is UQ = 0. Thus UQ(V) = UQ is constant. The equation for w on — oo < y < oo then becomes w" - w + w2 = 0, w'{0) = 0, w(0) > 0. (2.5) It follows that w = | sech 2 (y/2) as in (1.7). At next order, we obtain the following system: L 0 V i =. -r'0w' - ^JZlw> _ ^1 w h e r e L 0 V 1 = V7' - VX + 2wVu (2.6a) Using the self-adjointness property of LQ, the solvability condition for (2.6a) J W'LQVX = 0, 54 and therefore we have Note that U\ may be discontinuous at the origin. To evaluate the integral on the right hand side of (2.7), we obtain f^U^(U«-™\+U^)J^. (2.8) This is seen as follows. Integrating (2.6b) and using the even symmetry of w, we have: i ^ ( / o J » 2 + / o ^ 2 ) + ^ ( - H i / < o Thus Using the fact that w3(y) (fQy w2) is an odd function yields (2.8). This shows the claim. Finally, using (1.7) we have a 6 f 3 3 6 W =-, y«, =-. . This leads to the following O D E for the ring radius ro: r , + ^ = ^ l ( - ° ° ) + ^ l ( ° ° ) , (2.9) We now evaluate the right hand side by using the asymptotic matching with the outer solution. We obtain U0 = u(r0), U[(±oo)~u'(±r0) (2.10) and, N - 1  — 1 1 1 f urr(r) + ——Ur(r) - u{r) ~ - 1 + - - ^ w \ y ) u ( y ) + j^j-S(r - r 0 ) J where 6 is the delta function. Using J w2 = 6, we may then write: w2 ^ - l - ^ G ^ r o ) , (2.11) 55 where G is the Green's function satisfying N - 1 Grr + —-—Gr - G = -5(r - r 0 ) (2.12a) with boundary conditions Gr(0,ro)=0 = Gr(R,r0). (2.12b) It is easy to see that 1 [ J i ( r ) J 2 ( r 0 ) if r < r 0 G ( r , r 0 ) = z U (2.13a) W { Ji(r0)J2(r) if r 0 < r, where W is defined by W = J { ( r 0 ) J 2 ( r 0 ) - J i ( r 0 ) J 2 ( r 0 ) (2.13b) and where J i , J 2 satisfy TV" - 1 Jrr + —-—Jr - J = 0 with J{(0) = 0 and J'2{R) = 0. (2.13c) We write the solution to (2.13c) as J i ( r ) = J(r), J 2 ( r ) = i f ( r ) - ^ ^ / ( O , (2.13d) where J and i f satisfy (2.13c) but with R replaced by oo. Note that in the case N = 2, I,K are Bessel's functions of order 0. We may always scale J\ and J 2 so that N - 1 W = JUro) J 2 ( r 0 ) - J i ( r 0 ) J 2 ( r 0 ) = (2.14) In particular, when N = 2, (2.14) is satisfied with J\,J% as given in (2.13d). Wi th the above scaling for N > 2, we obtain G ( r 0 , r 0 ) = ^ - ^ ( r o j j ^ r o ) . (2.15) Therefore, from (2.10) and (2.11) we obtain 7 7 - 1 6 Ji{r0)J2(ro) , , UQ-1-AW0—w—'  ( 2 - 1 6 a ) ry((oo) + ^ ( - ^ (2.16b) - U ^ I*7"-") = - ^ f w h e r e , = (2.16c) t/o ^1(^0)^2(^0) ^0 Substituting (2.16c) into (2.9) we then obtain the following proposition: 56 Proposition 2.1.1 Let Q, = BR be the ball of radius R. In the low-feed regime A -C oo, the Gray-Scott model (2.2) admits a solution of the form V ~ AlT0W ' r = ^> r° = R ^ 2 T ) > * ~ Uo, for \r - r 0 | = 0(e) (2.17) where w is given by (1.7), UQ satisfies (2.16a) and ro satisfies the following ODE: d 2JN-1 (Ji(r0)J2(ro)n , 1 - t /p , 0 1 R , —r 0 = —e < V s—————.—— > where s = ———. (2.18) Here, J\,J2 are defined in (2.13). We now study the existence of equilibria of the O D E (2.18). We set r'0 = 0 in (2.18) and eliminate Uo using (2.16a). This yields / i 2 Rr>( N ( 1 _ 7 ) 2 r o (^1(^0)^2(^0))' cnm -v A = - e G C r c r o ) - ^ — where 7 = ^ _ t j l ( r o ) j a ( r o ) » ( 2 ' 1 9 a ) t/o = (2.19b) A graph of r 0 versus A2 for TV = 2 and i? = 5,00 is given in Figure 2.3. When i? = 5, the graph shows that A2 blows up as ro —> TR where r# = 3.943 is a root of 7. Moreover, A2 > 0 for ro G (0, r#) and all values of A are attained on that interval. On the other hand, when R = 00, A remains bounded for any ro, as shown on Figure 2.3. We show the following: Proposition 2.1.2 Suppose that R < 00, N > 2. Then there exists an rR € (0, R) which satisfies [Ji(rR)HrR)]'= 0. (2.20) Moreover, for any A > 0 with A = 0(1), there exists ro G (0,r#) and Uo G (0, 3) such that ^ i H ^ ) ' r = N ' u~u" (2-21) and ro, C/o satisfies (2.19). Suppose that R = 0 0 , N > 2. Then there exists Ac > VT2 such that for any A £ (0,:.4C) ; the solution to (2.19) with ro > 0, UQ G (0, ^) exists, and (2.21) holds. 57 Figure 2.3: Plot of ro versus A2, with R = 5 (solid curve) and R = oo (dashed curve). In the case R = 5, note the singularity at TR = 3.943. Proof. To show the existence of r# for finite R, note that ( J i J 2)'(.R) = J[(R)J2(R) > 0 whereas (JiJ 2 ) ' (0) = J i (0)4(0) < 0. Thus ( J i J 2 ) ' has a root in (0,R). Let rR be the first such root. Thus 7 < 0 for r 0 € (0, r^) . We next show that when A satisfies the conditions of the proposition, there exists a solution ro to (2.19a) with 7 < 0 at that point. There are two cases to consider. • Case R < 00 : Since 7 —> 0~ as ro —> r^ , we see that A2 blows up as ro —> r^ . Moreover, using (2.15) and (2.19a), it is easy to show that A —> 0 as ro —» 0. This shows the existence of solution ro € (0,r#) to (2.19a) for any given A > 0 when R is finite, and the corresponding value of 7 satisfies 7 < 0. • Case R = 00 : In this case, we first claim that 7 < 0 for all ro > 0. Since J\, J 2 > 0, this is equivalent to showing that u(r) = r ( J i ( r ) J 2 ( r ) ) ' is always negative. After some algebra, we obtain N - 1 u"(r) + u'(r) - 4u(r) = 2A^J x(r) J 2 ( r ) . (2.22) 58 Note that J i J 2 > 0, u(y) ~ -(N - l)C-^rf as y -> oo, and f -C, TV = 2 as y —> 0. { -{N -2)Cy2~N, N>2 In the equation above, C is some positive constant that may change from line to line. Thus u is negative on the boundary of an annulus {x : e < \x\ < R}, for any R big enough and any e small enough. It then follows from the positivity of J\ J 2 and the comparison principle that u is negative everywhere on that annulus. Since £ and R are arbitrary, u{r) < 0 for all r > 0. A simple calculation shows that for R = oo, . But then A2 —> 12 as y —> oo. This shows the existence of Ac > \ / l 2 , as well as the existence of r0 e (0, oo) satisfying (2.19a) whenever A € (0,AC). This establishes the existence of r 0 > 0 solving (2.19a) with 7 < 0. Next, note that 7 < 0 implies that ^ - j - G (0,1). Therefore (2.19b) admits a solution with UQ € (0,1). • 2.1.2 T h e b r e a k u p i n s t a b i l i t y i n t he low-feed r e g i m e In this section we study the breakup instabilities of stripe and ring solutions in the low-feed regime. The analysis here parallels the analysis of the large eignevalues in §1.1.2. First, consider the stripe solution on the rectangular domain Start with a one-dimensional single spike solution on x\ G [—1,1], as derived in Proposition 1.1.1, then extend it trivially in the x 2 direction. This yields the leading-order equilibria stripe solution in the inner region 7(7-) ~ - 1 + J V - 3 4 r 2 0 0 G(r, r) ~ 1, r —> 0 0 ft = [-1,1] x [0,d]. (2.23) 59 where U = U± is given by (1.11). We consider the following perturbations from the steady state: v = ve + cos(mx2)eA4<5f>(a;i), u — ue + c o s (mx2 ) e A V ( x i ) . (2.24) where, to satisfy the Neumann boundary condition, the mode m is an integer multiple of ir/d. We obtain the following linearized problem: \<j) = e24>" - m2£2(p -4> + 2Aveue<p + Av2tp, (2.25a) T\*/J = I/J" - m2*p - ^ (2iseue<f> + is2ip) . (2.25b) Near the core of the stripe, where x\ <C 1, we have « & ~ * ( y ) , vb{xx) ~ 0(0) = Vo, (2-26) 1 1 /" Aveue<fi ~ -veue4> ~ <5(xi)— / ~ "7772^Vo, - I / e V ' ~ ^ l ^ o 1 so that i>" - (1 + m2 - T\)VJ ~ 2<J(xi)-i ^ + o X x ^ o - ^ , (2-27) A$ = L 0 * + jjpw2Tpo (2.28) From (2.27) we obtain: f 2 f 'A2U2 Vo~Gm(0,0)|^yu,$ + 6 ^ 0 ^ l > . (2.29) where Gm is the Green's function satisfying GmXlXx - (1 + m2 + T\)Gm = -5(x[ - x x ) (2.30a) with Neumann boundary conditions at x i = ± 1 . We thus have G m (0 ,0 ) = coth(W), 6 = y/l + m2 + rA. (2.30b) 20 60 Combining (2.29) and (2.28) we obtain the following non-local eigenvalue problem: A$ = L 0 $ + X ™ 2 T 5 = °.  x = °2 i r ( 2 ' 3 1 ) J w Gm(0,0)s + 1 where s = and a is given by (1.10) with D = 1. As with Proposition 1.1.6, we therefore obtain the following instability result: Proposition 2.1.3 Suppose me -C 1. Let where U is given by (1.11). If s < sm then a single stripe solution given by (2.23) is unstable with respect to the mode m perturbation given by (2.24). Here, <f> is an even function. If s > sm and r < 1 then the mode m is stable. As shown numerically, an instability of this type leads to a breakup of the stripe into m spots. We therefore refer to this instability as a breakup instability of mode m. Since sm can be made arbitrary large by choosing large enough m, we have deduced the following Corollary 2.1.4 The stripe in a low-feed regime is always unstable with respect to breakup instabilities for large enough m. The analysis for a ring in two dimensions is the same, except a in (2.31) is replaced by GR(TQ, ro) as given by (2.13), and Gm(0,0) is replaced by GRm(ro,ro), where GRM is the radial Green's function of order m, satisfying s. V l + rn2 tanh ( /Vl + m2) tanh(Z) 1 - U U result: d2r GRm + --7-GRm TrGRm - (1 + r\)GRm = —6{r' r dr r -r). This yields GRm{ro,ro) = Jim(ro)J2m(ro)ro, 61 where Jlm(r) = Im(^r), J 2 m ( r ) = Km^r) - J m ( M r ) , /it = v T T r A , (2.32) where K m , I m are Bessel functions of order m. Thus we obtain the next result. Proposition 2.1.5 Consider the equilibrium ring solution of radius ro of Proposition 2.1.2, in two dimensions, and consider the breakup perturbation of the form u(r) + cos(mc?)eAV(r) (2.33) with (f> even, and me <C 0(1). Let J l ( r 0 ) J 2 ( r 0 ) s = 1 - Up . m Jim(ro)J2m{roY Uo where Jlmir) = Im(r), J2m(r) = Km(r) - ^fj^-Im(r), J, = J 1 0 , J 2 = J 2 0 , and Uo is given in (2.16a). Then the equilibrium solution ue is unstable with respect to the breakup mode m (2.33) if s < sm. Conversely, if s > sm and T « 1 then the mode m is stable. Note that Jim(ro)J2m(?o) = 0 ( l / m ) as m —> oo. Therefore, as with stripes, we have Corollary 2.1.6 The ring solution in a low-feed regime is always unstable with respect to breakup instabilities for large enough m. We remark that from our numerical computation, we find that in the case R = oo, the mode m = 1 is always unstable. Finally, we consider the case m = ^ with m 0 = 0(1). Then the linearized problem (2.25) for the stability of the stripe stripe becomes (A + m^)4> = e2(f>" -(f> + 2Aueue(f) + Au^ip, (2.34a) (2.34b) 62 It follows from (2.34b) that VJ ~ 0 in the inner region, and after changing variables as in (2.26) we obtain (A + m 2 , )* = L 0 $ , where LQ is defined in (1.16). From Lemma 1.1.4, LQ has a single positive eigenvalue | . Thus we obtain Re A < | — m 2,. The calculation for a ring is identical except that m 2 , gets replaced 2 by The result is summarized as follows: P r o p o s i t i o n 2.1.7 The stripe solution in the low-feed regime is stable with respect to breakup instabilities of mode m = ^ for all TUQ satisfying m\ > | • The ring solution of radius ro is stable with respect to breakup instabilities of mode m = ~ for all mo satisfying > f. Corollaries 2.1.4, 2.1.6 and Proposition 2.1.7 establish the existence of a wide instability band for the breakup instabilities in the low-feed regime, for either stripes or rings. The upper bound is m = O(^) and the lower bound is of 0(1). As we will see below, an instability band also exists for the intermediate and high-feed regime. 2.2 S t r ipe i n the Intermediate and high-feed regime In this section we discuss the stability of a stripe solution in high and intermediate regimes of the Gray Scott model in two dimensions 2.1. The equilibria stripe solution in the high-feed or intermediate regime is constructed by taking the one-dimensional pulse solution constructed in §1.2.1 and trivially extending it in the x2 direction. In this way we obtain the following equilibrium state: P r o p o s i t i o n 2.2.1 Let fl = [-1,1] x [0,d]. Let . B = tanh(i)A. Suppose A » 0(E1!2) and B < 1.347. Then there exists a stripe 'solution to (2.1) of the form v~^V0(y), u~^UQ(y), y = y , 63 where Vo(y), Uo(y) satisfy (1-42) with boundary conditions Vo(0) = 0 = f/0(0), Vo(oo) = 0, U'0{oo) = B. (2.35) In the intermediate regime B = 35 where e1/2 <C 5 <C 1, UQ and VQ are given in Proposition 1.2.2. 2.2.1 Zigzag instabilities of a stripe We now study the stability with respect to the small eigenvalues of the stripe equilibrium solution constructed in Proposition 2.2.1. The shape of these instabilities causes the stripe to develop a zigzag-type pattern in the transversal direction. We introduce a perturbation around the equilibrium state: v(x) = ±(v(y) + eXteim*><l>(y)), u(x) = j (u(y) + eXteimX2N(y)^ where y = y . Here, V,U satisfy (1.40), and m is an integer multiple of ir/d. The equations for $ , i V on —oo < y < oo become: A$ = - m2£2$ - $ + V2N + 2VU$, (2.36) T£2\N = N" - m2e2N - e2N - V2N - 2VU$. From (2.36), it is clear that there are several cases to consider, depending of the magnitude of m. First we consider the case m = 0(1). Then, for small eigenvalues, we expand $ = $0 +Ae$i +..., N = N0 + AeNi + ..., X = Ae\0 + The equations for $o,No and $i,Ni are then exactly the same as in §1.2.2 and are given by (1.84) and (1.85), respectively. Therefore the analysis of the inner region is exactly the same as in the proof of Theorem 1.2.4 up to (1.93). In particular we obtain as before <&o = Vo, No = UQ, and the mode m plays no effect there. For the outer region, we set v = ve(x\) + eXt cos(mx2)(/)(xi), u = ue(xi) + eXt cos(mx2)rj(xi), (2.37) 64 where ve,ue is the equilibrium stripe solution to (2.1), to obtain the following problem for the rj in the outer region: Vxx - (1 + rA + m2)rj = rjvl + 2ueve(f). (2.38) Comparing this with (1.94), we see that the only difference is that 1 + rA gets replaced by 1 + rA + rn2. Following the rest of the computations with this replacement, we have therefore shown the following result regarding the small eigenvalues for a stripe: Proposition 2.2.2 Consider a single stripe equilibrium solution (UQ, VQ), of (2.1) in the high- feed and intermediate regimes, as given by Propositions 1.2.1 and 1.2.2, respectively. There exists an instability the form U ~ UQ(y + cemX2eXt), V ~ V0(y + cemX2eXt), where m = 0(1), y = c is a small constant, and A is given by: A ~ —eAa (1 — 6 tanh 01 tanh I) , (2.39) where 6 = \Jl + Ar + m 2 , and a is a positive constant whose definition is given in Proposition 1.2.4.. Assuming r <C 0(j), (2.39) reduces to: A = -Aea-(l - \ / m 2 + 1 tanh (I) tanh(i V m 2 + 1)) . (2.40) Thus A is negative for small m but is positive if m is big enough. The threshold of stability occurs for m = mi, satisfying 1 - \/m2 + 1 tanh (I) tanh(Z Vra^+T) = 0. (241) The graph of I versus the first unstable mode mi is shown in Figure 2.4. In particular, the lowest unstable mode m/ is mi ~ 0 as I —* 0 0 . Additionally, it easy to see that mi ~ j as I —> 0 where z = 1.1997 is the unique root of z tanh z = 1. 65 20 18-3 16 14 12 771/10 8 6 4 2 i I Figure.2.4: Plot of the domain half-length / versus the lowest unstable mode m;. Above this, curve we have instability. Next, we consider the case 0(1) <C m < O(^). Then in the inner region, we expand the solution to (2.36) as A = meA 0 + . . . , $ = V0' + £ m $ i + . . . , N = UQ + emNi + We then obtain that \I>i, N\ satisfy the following system of -oo < y < oo : (2.42) X0V0' = -$! + V02Ni + 2 V 0 C / 0 * i , 0 = - iVxVb2 - 2V0Uo$i. (2.43a) (2.43b) To determine A 0 , we multiply (2.43a) by the adjoint solution which satisfies (1.88). By integrating over the domain, we obtain that J = . # t ( o o ) [N[{oo) + N[(-oo)} =X0J$\VQ'. (2.44) To determine A^i(±oo), we match the inner solution N with the outer solution 77, as in (1.96)- (1.97b). We have: 7?(X1) = E (U'o ( ^ ) + STTliVi Xl £ Xl 66 and also UQ(±OO) = 0. It follows that JV((±oo) = — r]'(±0). (2.45) em Note that 77 satisfies (2.38). Assuming that Ar <C 0 ( m 2 ) , we therefore obtain rj" — m?n ~ 0, x\ ^ 0 and 7/(0*) ~ ±eB = ±eAtanh(Z), r?'(±Z) = 0. Since m is large, we obtain that r? ~ . d 3 e - M H (2.46) Combining (2.45) and (2.46) we obtain N[(oo) + Nl(-oo) = -2B. Recalling (2.44), (2.42) and (1.102) we finally obtain A 0 ~ c M t a n h Z , (2.47a) A ~ ameA tanh I. (2.47b) This computation shows that in the high-feed regime, all of the modes m with O( l ) <C m <C O(^) are indeed unstable, whenever a > 0. As a remark, (2.47b) can be obtained formally by taking the limit m > 1 in (2.39). Equation (2.47b) is valid uniformly in the high-feed regime A = O ( l ) , for 1 <C m -C \ - However in the intermediate regime, this computation is not uniformly valid. This is because we have not included the term e 2 m 2 $o in the second order equations (2.43a). However as m is increased, the order of this term will eventually balance 0 (em$i ) . To see when this balance occurs, we formally rewrite (2.43a), but include the term e 2 m 2 $o = e 2 7 n 2 V 0 / : AOVQ' = &{ - meV0' - $1 + V^Nx + 2V0U0^1. 67 Proceeding as before, this leads to (2.47a) but with Ao replaced by Ao + me : A 0 + me ~ 01A tanh I. (2.48) Using (1.79), (1.50) this can be written as A ~ me(682 - me). (2.49) Thus (2.47a) is valid in the intermediate regime only when 0(1) -C m . < 0(^). However (2.48) is valid uniformly for O( l ) <§; m < O (JY) • Equation (2.48) also yields the upper threhold mu on the instability band: 6S2 , . m u ~ — . (2.50) Recall that 8 S> 0(e 1 ^ 2 ) ; thus we indeed have mu S> 1. Finally, we consider the case m = O ( i ) . We set 1 m = m o - e with mo = 0(1). The leading order linearized equations then become: (A + m 2 ) $ 0 = $o - $o + V02N0 + 2V0U0$Q, (2.51a) m20N0 = < - l/ 0 2iVo - 2VroL70*o- (2.51b) We first consider the intermediate regime. Using (1.62) we obtain: VQUQ = w + 82{yi + u\w) + ... VQ2 = 82w2 + 842wVl + ... Thus we expand: $o = $oo + ^ 2 $ o i + N0 = N00 + 82N0l + ... Since we are looking for odd solutions, we must also assume mo = Sfj, + ..., A = Ao<52 . . . . 68 The leading order equation for $01 then becomes: 0 = L 0 $oo, 0 = TVQ'O - 2w$00, where L0<& = + 2w$. Thus we obtain: $00 = w', NQO = u[- Equation for ̂ 01 a t next order is then ( / i 2 + X0)w' = L 0 $ o i + VJ2U[ + 2(vx + uiw)w'. ' (2.52) Multiplying by w', integrating, and using the self-adjointness of LQ, we obtain: (/J2 + A 0 ) J' w'2 = J w2u[w' + 2(vi + uiw)w'2. Next, we differentiate (1.62b) to obtain: LQV'I = —w2u'i — 2(i>i + u\w)w'. Multiplying this expression by w' and integrating, we obtain that the right hand side of (2.52) is zero. This yields A 0 ~ -112, A ~ - ( em) 2 . (2.53) This shows the stability of the higher modes m = 0 ( f ) . As a remark, note that (2.53) can also be obtained by taking a limit me S> <52 in (2.48). Thus (2.48) is valid uniformly for all O ( l ) -C m. Combining together (2.48) and (2.39) we can write a uniformly valid formula for the small eigenvalues in the intermediate regime: P r o p o s i t i o n 2.2.3 Consider a single stripe equilibrium solution (UQ{XX),VQ(X\)), of (2.1) in the intermediate regimes, as given by Proposition 1.2.2, and suppose that r < O ( l ) . Then the small eigenvalue A corresponding to the mode m as in Proposition 2.2.2 is given by: A ~ me { 652 ^^^tanhtx/^TTO C ° t h / — me m m (2.54) 69 Here, 5 is as in Proposition 1.2.2, and (2.54) is valid for all 0 < m < O (|) . This eigenvalue is unstable if and only if mi < A < mu, where mi is the root of the transcendental equation 1 = Jmf + ltanh(Z)tanh(ZWm 2 + 1), (2.55) and 6S2 The graph of I versus mi is shown on Figure 2.4- In the high-feed regime, the system (2.51) can only be solved numerically. To do so, we discretize (2.51) on a long interval [0, L] using centered finite differences, ensuring that $o and No are odd functions so that NQ(0) = $o(0) = 0. Choosing a meshsize h = L/n, where n > 1, we label yi = h and yn = L. This leads to the discrete eigenvalue problem (M - (1 + ml) I + A 2 ) $o + AiJVb = A*o , (M - m^I — A i ) N0 = A 2 * o , (2.56a) so that (M - ( l + m20) I + A 2 + Ax (M - m2QI - Ax) _ 1 A 2 ) * 0 = A $ 0 • (2.56b) Here 3>0 = (*o(yi), • • •, $o(2/n))', N0 = (N 0(2/i), • • •, iV 0 (y n ))*, and the matrices Ax, A 2 , and Ai, are defined by A l EE V02(yi) 0 0 0 0 \ VoHVn) J ( A 2 2U0(yi)Vo(yi) 0 0 0 0 \ 2tV0(j/n)Vb(3/n) J (2.56c) 70 and M = h? - 2 1 0 1 - 2 1 o '•. ••. o ••. ••. 0 0 0 0 0 0 0 0 '•• 0 0 '•• •• 0 '•• 0 1 - 2 1 0 2 - 2 (2.56d) Our computational results show that there is a critical value mo„ of mo for which Re(A) < 0 for mo > mou and Re(A) > 0 for mo < mo u- This re-stabilization value mo u is computed numerically from the discrete eigenvalue problem using L A P A C K [2] on a domain with L = 12 and n = 200 meshpoints. Increasing the number of meshpoints and the domain length did not change the results significantly. In Fig. 2.5 we plot the critical mode mo U ) corresponding to A = 0, for each point along the primary branch of the 7 versus B curve of Figure 1.6 (i.e. that part of the curve for which 7 > 1.02). Notice that the critical mode mo« tends to zero as B —-> 0. This agrees with our analysis of the re-stabilization value mu in the intermediate regime. We summarize our stability results obtained so far in the following statement: P r o p o s i t i o n 2.2.4 Using the notation of Proposition 2.2.2, suppose I > O ( l ) . Then there exists a band of unstable zigzag modes m satisfying mi < m < mu, where mi = O( l ) is given by by the root of (2.55) and mu 3> 1 depends on B = tanh(l)A. In the intermediate regime B = 36 <C 1 we have mu ~ In the high-feed regime, mu = r ! j l i - where mou depends only on B. Numerically, this dependence is given by the graph of Figure 2.5. \ As was mentioned after (2.41), m; is proportional to Z - 1 for small /. On the other hand, we 71 Figure 2.5: Plot of the re-stablization value mn u (with mu = e~lmou) versus B computed from (2.51) at each point along the primary branch of the 7 versus B bifurcation diagram. Above this curve we have stability with respect to zigzag perturbations. have B = ta.nh.lA ~ IA for small I. Thus when A = 0(1) and I is small we have B <C 1, which corresponds to the intermediate regime. We then obtain: 1.1997 1 2 , o , 0 mi —, m u ~ - - l A . I e 3 In addition, we require that the 0(e) extent of the core of the spike be much small than the domain size, so that e <C L Moreover, the intermediate regime requires the condition 52 ^> e. Equating mi = mu, we obtain the following condition for the stability with respect to all zigzag modes. Proposition 2.2.5 Using the notation of Proposition 2.2.2, suppose that I < 1, A2l2 » e, I > e (2.57a) and — < y , (2.57b) where z = 1.1997 is the unique root of z tanhz = 1. Then the stripe solution is stable with 72 respect to all zigzag instability modes m. In particular, (2.57) are satisfied when A = 0(1) and I3 < £ < I2. In our analysis, we have assumed that r < 0(1) as e —> 0. If we were to allow r = 0 ( e _ 1 ) , then for each fixed value of m we could get a zigzag instability due a Hopf bifurcation when r is increased past some threshold. We will not consider the case of asymptotically large r here. 2.2.2 B reakup Instabi l i t ies of a Str ipe In this section we study the stability with respect to the large eigenvalues of the stripe equilib- rium solution constructed in Proposition 2.2.1. Numerically, these instabilities are found to be the mechanism through which a stripe equilibrium breaks up into a sequence of spots. As in the study in §2.2.1 of the small eigenvalues, we look for a normal mode solution in the inner region x\ = 0(e) in the form v = I (V(y) + eXteimx^(y)) , u=j (u(y) + eXte™x*N{y)) , . (2.58) where y = e~lx\, U,V satisfy (1.40), and U, V, and N, have power series expansions in powers of eA. We now look for even functions $ and N. Assuming that r = 0(1), we obtain, as in §2.2.1, the following leading-order eigenvalue problem on —oo < y < oo: o A $ 0 = $ 0 ' - (1 + *o + V2N0 + 2V0U0$0 , (2.59a) 0 = NS - vNo - V2N0 - 2Vbt70*o • ' (2.59b) Here we have defined /J, by 2 2 fi = £ rn . To determine the instability band for (2.59), we discretise (2.59) on a. long interval [0, L] using centered finite differences, ensuring that $o and No are even functions and that NQV = 0 at y = L. The resulting matrix eigenvalue problem is similar to that in (2.56), except for slight differences in the matrix structure due to the different parity of the breakup eigenfunction. Our 73 computational results show that there are values m and pi2 for which Re(A) > 0 for /xi < \JL < fi2, and Re(A) < 0 for 0 < /i < m and \i > \x2. This instability band is determined from numerical computations of the discrete eigenvalue problem using L A P A C K [2] on a domain with L = 12 and n = 200 meshpoints. Increasing the number of meshpoints and the domain length did not change the results significantly. In Fig. 2.6 and in Table 2.1, we give numerical results for nx and \x2 versus B along the primary branch of the 7 versus B curve. 1.4 1.2 1.0 0.8 Ml, M2 0.6 0.4 0.2 0.0 0. Figure 2.6: Plot of fix (heavy solid curve) and /J,2 (solid curve) versus B computed from (2.59) at each point along the primary branch of the 7 versus B bifurcation diagram. In terms of m, the thresholds are rrij = ^/JlJ/e for j = 1,2. Although we are not able to calculate fj,\ and fi2 analytically in the high-feed-rate regime, we can asymptotically calculate these critical values, at which A = 0, in the intermediate regime where B = 35 « 1 with 0{e1/2) < 5 < O( l ) . In the intermediate regime we use Proposition 1.2.2 for Vo and UQ. In this way, (2.59) reduces to (H + A ) $ 0 ~ *o - *o + 2 (u; + 0(52)) $0 + (S2w2 + 0(64)) N0, (2.60a) fxN0 - - 2 (w + 0{52)) *o - (S2w2 + 0(64)) N0 . (2.60b) The balance of the various terms in the first equation gives rise to two possibilities: either O(N0)52 « O ( * 0 ) or O(AT 0)£ 2 = O ( $ 0 ) . In the former case, the leading-order equation for $ 0 T 1 1 1 1 r .0 0.2 0.4 "0.6 0.8 1.0 1.2 1.4 B 74 7 B Ml M2 1.464 0.40 0.003713 1.188 1.443 0.50 0.007010 1.154 1.419 0.60 0.01249 1.113 1.390 0.70 0.02093 1.069 1.357 0.80 0.03277 1.020 1.319 0.90 0.04796 0.9700 1.277 1.00 0.06588 0.9208 1.220 1.12 0.08830 0.8695 1.140 1.25 0.1096 0.8345 1.100 1.30 0.1156 0.8358 1.020 1.347 0.1617 0.8724 Table 2.1: The values of \x2 and /xi as defined in Proposition 2.2.6 along the primary branch of the 7 versus B bifurcation diagram. on —oo < y < o o is (/i + A)$ 0 = * o - * o + 2«;$o, ( 2- 6 1) with $o - • 0 as \y\ —* oo. From Lemma 1.1.4, this problem has a unique positive eigenvalue fi + A = f. This gives the upper bound of \i<i = | for the instability band. For the second possibility, we must expand N0 = ^ (iVoo + S2N0L + •••), /x = 5Vo + • • • , *o = *oo + £2$oi + ••••• (2.62) Substituting this expansion into (2.60), we get that NQQ satisfies NQ0 = 0. Therefore, iVoo is a constant to be determined. The leading-order equation for 3>o is A $ 0 0 = *oo - *oo + 2™$oo + A W 2 , (2-63) with $oo —• 0 as \y\ —> oo. To determine xVoo, we must match the inner solution to the outer solution. In the outer region, we have: v = ve(xi) + eXteimX2<f>(xi), u = ue(x{)-+eXteimX2r)(xi), . (2.64) 75 where ue,ve are the equilibrium solutions to (2.1), and EC EC n EC Substituting (2.64) into (2.1), we obtain the following eigenvalue problem on — I < xi < I: X4> = £ 2 < / > x i a : i - £2m24> - 0 + 2Aueve4> + Av\r\, (2.65a) W i - (1 + m 2 + rA) T? = 2ueve<f> + v2r], (2.65b) with r]xi (±1) = 0. The right-hand side of (2.65b) is localized near x\ = 0. Using Proposition 1.2.2 for solutions in the intermediate regime we obtain that 2ueve4> + v2r] ~ _ $ n o + _-AT 0 0 . (2.66) Therefore, we obtain the following problem for 77(2:1): V x i x i - (1 + m2 + rA) 77 = Q J w$oo dy+^f- J w2 dy^j 8(x{), -I < xx < I, (2.67) with rjxi(±l) = 0. The solution for 77 is ^1) = -[j J w$00 dy + ~ f ™2 dy^j Gm(xu0), (2.68) where Gm(xi,x[) is as given in (2.30). The matching condition for the inner and outer solutions is that j (5-2N00 + • • •) ~ 77(0) + • • • . (2.69) This yields that 77(0) = E6~2NQQ/A. Evaluating (2.68) at x\ = 0, we obtain an equation for the unknown constant iVno, i V o o ( ^ f ^ Jw2dy + ~ ^ = - ^ J w ^ 0 d y ) j G m ( 0 i 0 ) . (2.70) Solving (2.70) for A ôo, noting that J w2 dy — 6, and substituting the result into (2.63), we obtain the following nonlocal eigenvalue problem for $00 on —00 < y < co: $00 ~ $00 + 2vo$oo ~ X™2 1 = Afroo , (2-71) J wi dy 76 with $00 —> 0 as \y\ —* oo. Here x is defined by ^ 2 [ 1 + 6 ^ 0 ) ] " ' - < 2 - 7 2 ' Here, G m (0 ,0 ) is given in (2.30). From Lemma 1.1.3, we obtain the following sufficient condition for instability: . gx2 2tanh(Z\/ l + m2)Vl + m2 > —- . (2.73) This condition is also necessary provided that % is asymptotically independent of r ; i.e. when m2 > O(r) . But since 5 > 0 ( e 1 / 2 ) , (2.73) implies m » 1. By assumption r < 0(1), it then follows that (2.73) is also a necessary condition. Assuming I > 0(1), (2.73) then becomes m > . (2.74) £ This yields the lower limit of the breakup instability band in the intermediate regime. We summarize our result for breakup instabilities in the following statement: Proposition 2.2.6 Let e —» 0, r < 0(1) and I > 0(1). In the intermediate regime, the stripe equilibrium solution is unstable with respect to breakup instabilities if and only if the following inequality holds: 352 <Em<~. (2.75) In the high-feed-rate regime, the instability band along the primary solution branch of the 7 versus B curve is given by y^n < ™ < VT^2, ' ' (2.76). where fii = ^\(B) and \ii = ^(B) are given in Fig. 2.6 and in Table 2.1. 2.3 S tab i l i ty of a R i n g Solu t ion in intermediate and high-feed regimes In this section we modify the methods of §2.2 to obtain analogous results for the stability of an equilibrium ring solution for the Gray-Scott model (2.1) in the disk domain Q = {x : \x\ = r < R}. 77 2.3.1 R i n g radius and sp l i t t ing in the high-feed regime We first study the equilibria ring solutions in the intermediate and high-feed regime A 3> 0{ell2). The results of §1.2.1 concerning a pulse-splitting instability in the high-feed regime readily generalize to stripe or ring splitting. For a stripe, the threshold on A for the rectangular domain [—1,1] x [0, d] is the same as for a one dimensional spike splitting on a domain [—1,1], as given in Proposition 1.2.1. Similar analysis also applies for ring solutions, as we now show. In analogy with (1.39), we assume that the ring profile has the shape where both V, U are of order 1, and ro is the radius of the ring to be determined. We then obtain on — oo < y < oo that V V" + s V + V2U = 0, ro + £y 1 TT„ 1 U ' TT A V2U n ^u" + u + o - = o. e z . £ ro + £y £ £ Expanding as in (1.41), the leading term equations are then (1-42). Outside the core region of the ring, we have: FR u(r0) = l- G(r,rQ)v2(r)u(r)dr (2.77a) ujM^.k^fvSUa, (,77b) where G is the radial Green's function on the disk of radius R, given by (2.13). Since U(0) is of order 1, we obtain: / VO^ = ?TA y J-oo G ( r 0 , r 0 ) From (1.42b), this yields: 7J0(oo) - C/0(-oo.) = A G(r0,r0)' Normally, Uo will not be symmetric. However, on a disk of radius R, it is symmetric for the special case when ro = TR, i.e. when the following condition holds: (Ji(ro)Mro))'= 0, (2-78) 78 as we now show. From (2.77) we obtain: u ' ( r o) (G(r,rt))\r=rJ^V2(y)U0(y)dy. But the matching condition of the outer solution u and the inner solution U is: U'(±oo) = Au(r±). For UQ to be symmetric, we must have U'0(oo) + U'Q(—oo) = 0 which is equivalent to d 0 = ( G ( r , r 0 - ) ) | r = n ) + JL (G(r,r+))\r=ro = ±J[(rQ)J2(r0) + ^ J l (r 0) J 2(r 0) dr0 - u, 0 This shows (2.78). Note that ro given by (2.78) agrees with the limiting case A —> oo (see (2.19a)). Assuming that UQ is symmetric, we thus obtain the following boundary conditions for Uo and U^O) = 0 = V"(0), r/£(oo) = - — ^ Vb-> 0 as y ^ oo. (2.79) 2GR(rR,rR) As was argued in §1,2.1, the solution to the boundary problem (2.79) and (1.42) exists if and only if UQ(OO) < 1.35. We therefore obtain the following result: Proposition 2.3.1 LetrR be a root of (Ji(rR)J2(rR))' = 0, (2.80) and let B = * y (2.81) 2G(rR,rR) Suppose that 0(£1'2) < B.< 1.35, (2.82) 79 where J\,J2 are given by (2.13d) and G is given by (2.13). Then there exists an equilibrium ring solution to (2.1) given by < z ) ~ - V 0 ( y ) , u ( * ) ~ £ ^ M , y = f c ^ , ' (2.83) £ A £ where UQ, VQ satisfy (1.^2) with the boundary condition (2.79). Numerically, when B is given by (2.81) is just above 1.35, ring splitting is observed (see §2.4). Next we look at the zigzag instabilities of a ring. Expand V and U as in (1.41) to obtain the following equations for U\, V\ on — oo < y < oo: V VI" - Vi + IVQUQVi + V02U1 = - - ^ , (2.84a) Ar0 E/{'+ 1 - 2Vot/oVi - V ^ U ^ = ~ A \ - ( 2 - 8 4 b ) The asymptotic boundary conditions for U\ as y —> ±oo are to be obtained by matching. Next, we introduce a perturbation around the equilibrium state in the form v = ve(r) + eXteim0<j>(r), u = ue(r) + eXteimdri(r), (2.85) where f> <C 1, r\ <C 1, and m is a non-negative integer. Substituting (2.85) into (2.1), we obtain, in place of (2.65), the following eigenvalue problem on 0 < r < R: ( 1 \ £ 2 m 2 \f> = £2 ( far + -4>r J Y~$ _ ^ + 2Aueve4> + Av2-n, (2.86a) 1 m 2 rXn = r]rr + ~rjr - ~rj - T? - 2ueve<f> - v\r]. (2.86b) We will study this problem in the inner region where y = e 1(r — ro) = 0(1), and in the outer region where r — ro = 0(1)- In the inner region, we write „ e ( r ) = ^ M , ue(r) = jU(y), <f>(r) = U(y), n(r) = jN(y), y = £~\r - r0). (2.87) 80 In terms of these variables, (2.86) reduces to an eigenvalue problem on — oo < y < oo' 2 2 A$ = $" + — - $ ' - £ m $ _ $ + V 2 AT + 2Vc7$ , (2.88a) r 0 + £y (r 0 + £y) 2 2 2 T£ 2AJV = JV" + —-—N' - . £ M . - J V - e2xV - F2AT - 2V£/$ . (2.88b) r 0 + £y (r 0 + £y) 2 We first assume that m = O( l ) as £ —» 0. B y expanding V , {/, $, and N, in powers of e.A, and writing A = £^4Ao, we derive (1.84) for $o and No. Thus, $o = V0' and No = U'0. The system for $ i and N\ is (1.85), with the terms — &Q/(Aro) and —N'0/(Aro) added to the right-hand sides of (1.85a) nd (1.85b), respectively. Since these additional terms are even functions, they do not contribute to the solvability condition that determines Ao- Therefore, the entire analysis of (1.87-1.89) can be repeated, and we obtain that Ao satisfies Here for j = 1,2 is the solution to the adjoint problem (1.88). To determine N[(±oo) we must consider the outer region. B y repeating the analysis of (1.94)- (1.96), we obtain that the outer solution for 77 on 0 < r < R satisfies 1 TTI.2 Vrr + -VT n"?? ~ (1 + TA)T7 ~ 0, T ^ r0 (2.90) with rjr(R) = 0 and 77 r(0) = 0. The matching condition of the inner and outer solutions for rj is j (No + eANx + • • •) ~ 77 (4) + eyvr ( r f ) + ••• • (2-91) Since Ao = U'Q satisfies Ao(±oo) = ± B , we get from this matching condition that where G(r,r') is the radial Green's function satisfying GTT + ^GT — (1 + T\)G = —5(r — r'). Solving (2.90) for rj(r), we obtain that , V (ro~) J%m(r) / J2,m(ro), r<r0<R, r](r) = <( V (ro ) J\,m(r) / J\,m(ro), 0 < r < r o . 81 Here J i , m and J2<m are denned in (2.32). Using (2.92) and the Wronskian relation Jl,m(r0)J2,m(r0) - Jl,m(ro)J'2m{r0) = — , we obtain the compact formula Ni r((+oo) + N[(-oo) 1 2 Ar^Ji(ro)J2(ro)Jltm(ro)J2im(ro) ' (2.93) where J\ = J\fi,J2 = J2fl. Substituting (2.93) into (2.89), and recalling that A ~ eAXo, we obtain the following result for A: Proposition 2.3.2 Consider an equilibrium ring solution of radius ro for the Gray-Scott model in a disk of radius R as constructed in Proposition 2.3.1. Suppose that m = O(l ) as £ —> 0. Then, for a perturbation of the form (2.85), A satisfies the transcendental equation The constant a is as in Theorem 1.2.4 and is positive along the primary branch of the 7 versus B curve. The functions JiiTn, J2,m ar^ defined in (2.32) and J\ = J\$, J2 = J2fi- We will only consider the case where r = 0(1) as £ —* 0. Since A = O(e), we get 9 ~ 1. Therefore, from (2.94) we conclude that A is positive, and hence we have a zigzag instability, if and only if Using the asymptotic expansion for large m and fixed R, it is easy to show that JitmJ2tm — O(^) for m 3> 1 and ro fixed (see below). Therefore, from (2.95), we have an instability if m is large enough. In Fig. 2.7 we show the numerical computation of the first unstable mode m = mi versus the ring radius ro- The corresponding disk radius R can be determined in terms of ro from (2.78). From this figure we notice that the first two modes m = 1, 2 are stable for any ro. Also note that the first unstable mode increases as the ring radius ro is increased. This is in contrast to the result for stripes in §2.2.1, where the first unstable mode tends to zero as the domain half-length I is increased. (2.94) ^rlJi{r0)J2(ro)Jitm{ro)J2,m{ro) < 1. (2.95) 82 Figure 2.7: The graph of the first unstable mode mi versus the ring radius ro when r = O( l ) . The dotted line is the asymptotic curve (2.99). We cannot analytically determine m; for arbitrary R. However, by considering the two limiting cases R <C 1 and R 3> 1, we can obtain the following limit results: Proposition 2.3.3 The following asymptotic, formulae relate the domain radius R and the equilibrium ring radius ro, as defined by the solution to (2.78): R ~ V2r0 , R -> 0 . R ~ ro + ^ ln(2r 0 ) , .R-+oo. (2.96) (2.97) Suppose that T = ' 0(1), and let mi be the smallest value of m for which the eigenvalue of Proposition 2.3.2 is unstable. Then mi = 3 , R —> 0, mi ~ \/2ro , R —> oo . (2.98) (2.99) By calculating ro in terms of R from (2.97), (2.99) can be written as m ~ J2(R - \ ln(2i?) + •••)• We now derive this result. We first consider the case R —> 0. A simple calculation using (2.13d) shows that the equilibrium condition (2.78) for ro can be written as Ko{ro) K'0(R) _ 1 Jo(r0) i'o(R) 2r0Io(ro)l'0(ro) ' (2.100) 83 For R —> 0, with 0 < ro < R, we use the small argument expansions of IQ and Ko to obtain that 2 r 0 / 0 ( r 0 ) / 0 ( ro ) ~ r 2 , ^ § ~ - 2 / T 2 , ^ ~ - Inr . r o( f l ) " W o ) Therefore, (2.100) reduces to 2R~2 ~ r " 2 . This yields i? ~ v^ro, which establishes (2.96). Next, we determine the stability threshold ra; for R —• 0. Using the small argument expansions of IQ and i-fo, we readily obtain that Ji(r0)J2(r0) ~ - r o 2 . (2.101) Using (2.32), and the small argument expansions /Mn,)~ir( m ) (2)~ im[rt>) ~ _ L _ ( |)- , (2.102) we calculate for i? -C 1 and r « 1 that Jl ,m(ro) J 2 ,m(ro) = Im(r0)Km(r0) - ( 7 m ( r 0 ) ) 2 ~ ( l + )̂ . (2.103) Here r(ra) is the Gamma function. Therefore, substituting (2.101) and (2.103) into (2.95), we have instability when R <C 1 if and only if m - 2 > 0 . (2.104) •1 + 2 ~ m The first integer for which (2.104) holds is ra; = 3, which establishes (2.98). Next, we derive the results in Proposition 2.3:3 for R S> 1. To do so, we need the following standard asymptotic formulae (cf. [1]): * ° < W £ e ~ T ^ ) ' M ~ y ^ v + h ) ; ( 2 1 0 5 ) Therefore, we get KQ(ro) rr / 1 \ K'Q(R) r r / 3 \ , e W 1 i o M ^ r ^ j ' 4iU' J o ( r ° ) / o ( r o ) ~ 2 ^ v ^ Substituting these formulae into (2.100), we get that - ^ - ^ - T r e W l - A 2r 0 V 4i? 84 Solving this equation asymptotically, we get R ~ ro + \ ln(2r 0) as R —> co, which establishes (2.97). Finally, we establish the stability threshold (2.99) for R » 1. Using (2.105) and (2.97), we readily estimate that Ji(ro) J 2 ( r 0 ) ~ \ (1 + e2r°-2R) ~ ^ ( l + ^ ) . (2.106) Next, we must calculate Ji,m(ro) J2,m(ro)- Since, we have m/ 3> 1 and ro S> 1, we must use the following uniform expansions of Km(mz) and Im(mz) for m -> oo as given in [1]: / TT- e~mP(z) i em/3(z) Km(mz) ~ ̂ 7 1 + ^ 1 7 1 . ~ ^ ( i + , 2 ) 1 / 4 > ( 2 - 1 0 7 ) where B(z) is defined by /?(*) = v i + ^ 2 - f - l n ^ - l n ( ^ l + \ / r + ^ 2 J . (2.108) Defining z and z\ by r 0 2 = — , z i = — m . . m we get using (2.107) that Jl,m(r0) J 2 , m ( r 0 ) ~ o i 7 1 + e 2 - ^ ) - ^ ) ] ) . (2.109) 2m V I + z v y bmce i? = r 0 + ± l n ( 2 r 0 ) , we have that z — z\ —> 0 provided that ln(ro)/m < C 1. Assuming for the moment that this condition is true, we can then use B\z) = z~l\/l + z2 to estimate 1 B(z) - 3(Zl) ~ B'(z)(z - zx) ~ ln (2r 0 ) \ / l + 2 2 . Substituting this expression into (2.109), and using (2.106), the stability threshold condition defined by 4rg J i(r 0 ) J 2 ( r 0 ) J i ] T n (r 0 ) J2,m(ro) = 1 becomes i1 + 2ro") i1 + e _ l n ( 2 r o ) z ~ l v / I + ^ ) ~ 1, (2.110) v T + i 2 where z = ro/m. If we assume that z = 0(1) and r » l , then it is easy to see that there is no root to (2.110). Therefore, we must assume that z » 1. In this limit, (2.110) reduces to VT^z2 85 Squaring both sides, we readily obtain that r 0 ~ 2z2. Since z = r0/m, we get that m = v /2ro, which establishes (2.99). The consistency condition m(2r 0 ) /m <C 1 needed above is indeed satisfied. The upper instability threshold in estimated in the same way as was done for a stripe in §2.2.1. The only difference is that m gets replaced by ̂ . Therefore, we obtain the following results for intermediate and high-feed regimes: Proposition 2.3.4 Consider an equilibrium ring solution of radius ro for the Gray-Scott model in a disk of radius R as constructed in Proposition 2.3.1. Then, all zigzag perturbations of the form (2.85) are unstable in the zone mi < m < mu, and are stable outside of this zone. Here mi is the root of Ar2. Ji(ro)J2( ro) Ji,m(ro)J2,m(ro) = 1; which depends on R. In the high-feed .case A = 0(1), we have mu = m^T° where the threshold mo„, plotted in Fig. 2.5, is independent of R but depends on the value of B along the primary branch of the 7 versus B bifurcation diagram. In the intermediate regime B = 36 with 0(e1/2) C<5c O(l), mu is determined explicitly as 6r062 mu = . £ As with the stripe, we find that the instability band can dissapear in the intermediate regime for a small domain size. As found in Proposition 2.3.3, m; = 3 when 1? « 1. On the other hand, assuming ro -C 1, A < 0(1), and using (2.81), (2.13) and (2.106) we obtain: 35 = B = r 0 | , r 3 oA2 6£ On the other hand we must have £ < R = O(r 0 ) . In addition, the intermediate regime must satisfy 6 > 0 (e x / 2 ) . This yields the following stability result: Proposition 2.3.5 Suppose R -C 1. Then the equilibrium ring solution in the intermediate regime is stable with respect to zigzag instabilities provided that rlA2 < 18e < r 0 . and 0{£ll2) < r0A < O ( l ) . 86 In particular, the zigzag instabilties are stable provided that A = 0(1) and 0(R3) < e <C 0(R2). Here, ro ~ is the ring radius as derived in Proposition 2.3.3. Finally, the stability property of a ring with respect to a breakup instability is very similar to that of a stripe. We have, Proposition 2.3.6 Consider an equilibrium ring solution given in Proposition 2.1.1, in the in- termediate regime B = 35 <C 1. Such a solution is unstable with respect to the large eigenvalues, if and only if the following inequality holds: 3S2r0 <em< - ^ - r 0 . (2.111) In the high-feed-rate regime, the instability band is given by yfjTiro < sm < y/Jj^ro , (2.112) where fi\ = m(B) and fi2 = ^(B) are given in Table 2.1 and plotted in Fig. 2.6. The result (2.112) follows immediately by noting that when mo = me = 0(e), the leading-order problem for (2.88) become: 2 (A + ^ ) $ 0 = ${,' - $ 0 + V2N0 + 2V 0 tV 0 $o, (2.113a) 2 ^ J V 0 = N'o' - VQ2No - 2Vb(7o$o . (2.113b) ro But these equations are the same as (2.59) upon relabelling em by The result (2.111) is also readily obtained by a slight modification of the analysis in §4. To obtain (2.111), we repeat the analysis of (2.60)-(2.72). The upper bound in (2.111) follows from (2.61). Then, to determine the lower bound in (2.111), we re-derive the nonlocal eigenvalue problem (2.71), where the multiplier x m (2.72) is to be replaced with Gm{ro\ro) =.roJitm{ra)J%m(ro) • (2.114) X = 2 1 + 8 6 6G m ( r - 0 ; r 0 ) 87 Here G m ( ro; ro) is the Green's function, evaluated at r = ro, for the operator 77" + r~lr]' — m2r~2ri = —S(r — ro) with ^'(0) = n'(R) = 0. In (2.114), J i > m and J 2 , m are defined in (2.32). From Lemma 1.1.3, the stability threshold occurs when x = 1- Since e<5~2 -C 1 in the intermediate regime where 5 3> 0(e 1 / / 2 ) , we conclude that all modes with m = O( l ) are stable. However, since Jitm.J2,m ~ ( 2 m ) - 1 as m —> oo for fixed ro, we have that x w d l decrease below unity when m is below the lower bound in (2.111). 2.4 N u m e r i c a l Examples In this section we give a few numerical examples to support the results of this chapter. For a rectangular domain, the computations have been done with the software package " V L U G R " [6]. For a disk domain, we wrote our own code. We used a 2nd order finite difference uniform discretization in r and 6, combined with the forward Euler method in time. Matlab was used for visualization. E x p e r i m e n t 1: r i n g breakup and sp l i t t ing instabi l i t ies . For this experiment we chose a disk domain of radius R = 3, with e — 0.05, r = 1, and discretized the radial and anglular direction into 60 and 30 intervals, respectively. The time step was taken to be 0.00005. For initial conditions, we chose a ring of radius ro = 1.5 of width e, and with very small, random perturbations in the angular direction. Note from Proposition 2.3.1 we expect the ring to have the equilibrium radius of TR = 2.238. Also, from Proposition 2.3.1, the equilibrium exists only when A < 1.837. Figure 2.8 shows a simulation for A = 2.0. The initial ring at ro = 1.5 starts to expand until its radius reaches about 2.25. It then breaks into many spots. This implies that all lower-modes are stable', but a very high instability mode is triggered. Moreover, the spots form both at the outside and at the inside of the ring - thus the splitting instability is triggered at about the same time as the breakup instability. The ring breakup occurs at about r = 2.25 which is close to the equilibrium ring radius TR = 2.238. In Figure 2.9 we take A = 2.5. In this case the ring first splits into two. The two resulting 88 t=20, A=2.0 t=60, A=2.0 t=70, A=2.0 Figure 2.8: Contour plot of v for A = 2.0. t=10, A=2.5 t=20, A=2.5 t=80, A=2.5 t=340, A=2.5 t=480, A=2.5 t=600, A=2.5 Figure 2.9: Contour plot of v for A = 2.5. 89 t=0, A=4.0 t=20, A=4.0 t=60, A=4.0 Figure 2.10: Contour plot of v for A = 4.0. rings then start travelling apart. Some time later, the inner ring breaks up. Then much later the outer ring also breaks. Finally, in Figure 2.10 we take A = 4. As a result, a single ring eventually splits into four. The resulting rings then lose their stability, one-by-one, starting from the innermost ring, and progressing towards the outermost ring. Note however that the outer ring can remain stable for a very long time, and becomes unstable only after the adjacent ring has been broken up. E x a m p l e 1: Z igzag Ins tab i l i ty of a S t r ipe : In order to observe zigzag instabilities, one must choose parameters and initial conditions carefully so that these instabilities are not dominated by breakup instabilities that occur on shorter timescales. In this example, we consider a stripe equilibrium with e = 0.004, A = 1.7, and r = 1.0, in the rectangular domain [—1,1] x [0,1] so that d = I = 1. For this value of A, we compute that B = A tanh 1 = 1.3 < 1.35. Hence, by Proposition 1.2.1, there is a stripe equilibrium solution centered along x\ = 0. For these parameter values, we obtain from Proposition 2.2.2 that the unstable modes m satisfy rrib < m < m0u/£, where m; « 1.05 when I = 1, and mou « 0.72 when B = 1.3. This yields the 90 instability zone 1.05 < m < 180 when e = .004. The meshsize in the x\ direction must be finer than e in order to resolve the stripe. Wi th e = 0.004, the computation would be quite prohibitive for most uniform-grid codes. Fortunately, V L U G R uses an adaptive mesh-refinement algorithm, which captures the localized structure of the stripe without the need for a huge number of grid points. For the initial conditions for (2.1) we took v = jw —cos(i07rx 2) j a n c j u _ £^ w n e r e w(y) = |sech 2(y/2) satisfies (1.7). Therefore, the preference for a zigzag instability corre- sponding to the mode m = 107T, which lies within the instability zone, is built in. The resulting numerical simulation is shown in Fig. 2.11. The system does indeed develop a zigzag instability corresponding to this mode. t=6.6 t=13.6 -0.05 0 0.05 -0.05 0 0.05 Figure 2.11: Contour plot of v obtained from the full numerical solution of (2.1). The parameters are £ = 0.004, A =• 1.7, r = 1.0, and the domain size was set to [—1,1] x [0,1]. The initial condition was taken to be v = ^ X l + 1°—cos(27r5x2)^ u = e, where w(y) = |sech 2 (y/2). Outside the region shown, v is exponentially small. Numerically, we can also validate the theoretically predicted form of the zigzag instability. From 91 the theory of §3, we would expect that the eigenfunction </> has the form d 4> = ——-ve(x\) cos mx2 , (2.115) dxi where ve is the equilibrium stripe. Since v ~ ve + CeXtcj) and A = 0(e) 1, we have that <j> is well-approximated by the difference in the numerical solution at two different times. From results obtained from our numerical simulations, we plot vt=22 — ^t=i8 m Fig- 2.12. From this figure we observe that the shape of the resulting perturbation is indeed of the form (2.43a) V(t=22)-V(t=18) V(t=22, x2=0.5)-V(t=18, x2=0.5) • -0.05 0 0.05 -0.1 -0.05 0 0.05 0.1 Figure 2.12: Left: contour plot of 'ut=22 — i>t=i8 f ° r the data of Fig. 2.11. Right: The horizontal slice of the figure on the left at x2 = 0.5. / The onset of a breakup instability for this example is visible in Fig. 2.11 at time t = 32, when the wiggled stripe starts to develop an instability. Shortly after this time, the stripe breaks up into spots, which then self-replicate until the entire rectangle is full of spots. E x a m p l e 2: B r e a k u p Ins tab i l i ty of a S t r ipe : Next, we verify Proposition 2.2.6 for the parameter set e = 0.004, A = 1.313, I = 1, and d = 2. For this value of A, we compute that B = yltanh(l) = 1. From Table 2.1, we have have ^ i = 0.0659, /*2 = 0.9208. Therefore, from Proposition 2.2.6, the instability zone is 64.2 < m < 239.9. To check this, we start with the following initial condition for v that has a built-in preference for the the mode m = 20TT = 62.8: v=\(X + 10" 4 cos(207r:c2))™ ) , u = e, (2.116) where w(y) = |sech 2(y/2) is given by (1.7). Note that this mode is stable. Indeed, in the 92 t=44 -1 -0.5 t=51 0.5 t=67 2 ... 2 1.5 j 1.5 1 j 1 0.5 j 0.5 0 • 0 -1 -0.5 0 0.5 1 Figure 2.13: Contour plot of v computed from the full numerical simulation of (2.1). The domain size was set to [—1,1] x [0,1], and the parameter values are e = 0.004, A = 1.313, and r = 1.0. The initial condition was taken to be (2.116). resulting simulation shown in Fig. 2.13, the stripe breaks up into 21 and not 20 spots. This corresponds to m = 21TT = 66, which lies just within the theoretical instability band. E x a m p l e 3: Space -F i l l i ng C u r v e : In this example, we numerically show the development of a zigzag instability in the weak interaction regime, where both u and v are localized near x\ = 0. For convenience, we introduce an extra parameter D to be the diffusivity coefficient of u. In this way, (2.1) is replaced by vt = e2Av — v + Av2u, rut = DAu — u + 1 — v2u. (2.117) For this computation we take i — 0.05, D — 0.01, A = 2.0, I = 1, and d = 5. This corresponds to the weak interaction regime, whereby both u and v are localized near the stripe, and is similar to parameter regimes studied in [63], [64], and [72]. Notice that by re-scaling, these parameter values are equivalent to taking e = 0.5 D = 1, I = 10, and d = 50, in (2.1). Therefore, since e is 93 r a t h e r la rge , t h e t h e o r e t i c a l a n a l y s i s o f §3 a n d §4 is n o t a p p l i c a b l e . T h e n u m e r i c a l r e su l t s a re s h o w n i n F i g . 2.14 a n d F i g . 2 .15. T h e z i g z a g i n s t a b i l i t y c o r r e s p o n d i n g t o t h e m o d e cos (2^52:2) b e c o m e s u n s t a b l e a n d , i n c o n t r a s t to E x a m p l e 2, we d o n o t o b s e r v e a n y i n s t a b i l i t i e s t h a t l e a d t o t h e d e v e l o p m e n t o f s p o t s . T h e e n d s t a t e seems t o b e a d o m a i n - f i l l i n g c u r v e . W e r a n t h e s i m u l a t i o n to a b o u t t = 10000; t h e s o l u t i o n seems to have r e a c h e d a s t e a d y s t a t e at t h i s t i m e , s ince no changes i n the s o l u t i o n were o b s e r v e d b e t w e e n t = 8000 a n d t = 10000. t= 917 t=2137 t=2346 t= 4933 F i g u r e 2.14: C o n t o u r p l o t o f v for E x a m p l e 3 at t h e t i m e s i n d i c a t e d . Example 4: Zigzag Instability of a Ring: I n t h i s e x a m p l e , we c o n s i d e r (2 .117) a n d we t a k e t h e p a r a m e t e r va lues e = 0 .05, D = 0 .01 , A = 2.0, i n a r e c t a n g u l a r d o m a i n O = [—2, 2] x [0,4] , so t h a t I = 2 a n d d = 4. S i n c e t h i s p a r a m e t e r set a g a i n c o r r e s p o n d s t o t h e w e a k i n t e r a c t i o n r eg ime , i t is n o t i n t h e s c o p e of o u r t h e o r e t i c a l a n a l y s i s . W e t a k e t h e i n i t i a l c o n d i t i o n t o b e a s p o t at t h e cen te r o f the d o m a i n . A f t e r a s h o r t t i m e , t h e s p o t g r o w s i n t o a n e x p a n d i n g r i n g . S o m e t i m e l a t e r , t h i s r i n g deve lops a z i g z a g i n s t a b i l i t y c o r r e s p o n d i n g t o t h e m o d e m = 4. T h e e n d s t a te of t h i s c o m p u t a t i o n is a g a i n a c o m p l i c a t e d d o m a i n - f i l l i n g c u r v e as s h o w n i n F i g . 2 .16. 2.5 Discuss ion T h e re su l t s o f § 2 . 1 were p r e v i o u s l y p re sen ted i n [48]. I n a r e l a t e d w o r k [56], M o r g a n a n d K a p e r have a lso p e r f o r m e d a s i m i l a r a n a l y s i s o f r i n g - l i k e s o l u t i o n s o n G r a y - S c o t t m o d e l . B e l o w w e r e v i e w a n d c o m p a r e t h e i r r e su l t s w i t h ou r s . T h e y u s e d t h e f o l l o w i n g s c a l i n g o f t h e G r a y - S c o t t 94 - 1 -0.8 -0.6 -0 .4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 2.15: Plots of v(xi,x2) versus x\ with x2 = 2.5. model: dV(y,s) = DmkAV -BV + UV2 as dU(y, s) ds By scaling the variables as follows: = AU + Amk(l - U) - UV2 y = V AmkV, U = u, s = -jU, y = 2. X we re-obtain our system (2.1) with or ^ m k ~ r2A2" ^ = VA2' ®mk ~ e%T' The paper [56] obtains results for the ring location and its stability. In addition, Morgan and Kaper also perform a linear Turing analysis for the radially-symmetric solutions and full numerical simulations. 95 t=424 t=689 t= 917 t= 1947 t= 4098 t= 17984- Figure 2.16: Contour plot of v for Example 4 showing the zigzag instability of a ring and the labyrinthine pattern that results at later time. For the location of the ring, Morgan and Kaper obtain the same formula (equation (2.35) of [56]) as we obtain in Proposition 2.1.2 (see also (2.19)). However, they do not have any analytical results on the existence of solution to equation (2.35) of [56]. Indeed, they consider only a bounded domain - in which case a ring solution exists for any choice of A. Our result on existence of the bound Ac on A in case of the unbounded domain is new. In the case of the boudned domain, we rigorously show the existence of rR < R, which has the property that the radius of the ring ro —* rR as A —> oo. This is also a new result. Another new result that we have obtained using the comparison principle, is the existence of A for any given ring radius ro in the case of the unbounded domain. For the stability analysis with respect to angular perturbations, we obtain a simple sufficient condition (see Proposition 2.1.3) for when the ring is unstable with respect to node m. This condition is also necessary when r = 0. Our condition involves only Bessel functions of order m and the ring radius ro- Our proof involves no numerical computations. B y contrast, the stability criteria for the m-th mode in [56] is implicitly contained in integrals of hypergeometric 96 functions which is then solved by mathematica. In both cases, the stability analysis is reduced to a non-local eigenvalue problem. We also find that all nodes 1 « m « O(^) are unstable for any r < 0(1), and all nodes m > a r e stable (Proposition 2.1.7). These results are new. For the ring-splitting regime A = O( l ) , we use formal asymptotics and one-dimensional nu- merics to derive an explicit bound Ac in terms of R (see Proposition 2.3.1) such that the ring-splitting occurs when A > Ac. This is a new result. We then use full numerical simulation to confirm existence of the ring-splitting regime. Morgan and Kaper also observe ring splitting numerically, but they do not perform any analysis of this regime. We have analyzed zigzag and breakup instabilities of stripes and rings for the Gray-Scott model (2.1). For the breakup instabilities, we have established the presence of instability band in either low-feed or high-feed regimes. For the zigzag instabilities, we have analysed only intermediate and high-feed regimes. In the intermediate regime, we have shown that it is possible for a stripe or a ring to be stable with respect to all zigzag modes, when the domain is small enough (with D = 1)- Altenatively, this is equivalent to fixing the domain size and taking the limit D 1. In particular, we expect the stripes to be stable with respect to zigzag instabilities in the well-known shadow limit D —> oo. A natural extension of this work is to determine whether there is a parameter regime where the breakup instability band disappears. One way to stabilize a stripe is to take the domain width d to be very small. Indeed, since the upper bound of the instability band is 0 ( e _ 1 ) , a stripe can be stable if we take the domain width to be of order O(e). Such an analysis was performed in [21]. A different possibility is to analyze zigzag and breakup instabilities of a stripe and ring solution in the weak interaction regime of [63], [64], [72], and [35], where u and v are both exponentially localized near a stripe or a ring. This regime corresponds to taking Dv = 0(DU) < 1 in (7). The numerical computations of Examples 3 and 4 of §6 indicate that there exists values of D = Dv/Du = 0(1), with Dv = 0(DU) <C 1, so that a stripe or ring is stable with respect to breakup instabilities, but is unstable with respect to zigzag instabilities. We conjecture that in the absence of such breakup instabilities a zigzag instability leads to a 97 labyrinthine pattern such as shown in §2.4. A n open problem is to study zigzag and breakup instabilities of stripes and rings for the Gray-Scott model in the weak interaction regime. 98 Chapter 3 Gierer-Meinhard model in two dimensions In this chapter we concentrate on the study of the Gierer-Meinhardt (GM) model [26] in two dimensions. In its dimensionless form, the system is: Ap At = e2 AA - A + — , a: eft, t > 0 , (3.1a) Ar rHt = DAH - H + - , a; G ft, * > 0, , (3,1b) dnA = dnH = 0, xedQ. (3.1c) Here ft is a bounded two-dimensional domain, A and H represent the activator and the inhibitor concentrations, e 2 and D represent the diffusivity of the activator and inhibitor, r is the inhibitor time constant, and the exponents (p, q, r, s) satisfy ' P > 1 , q>0, r > 0 , S > 0 , IZ1<-!—. (3.2) q s+1 As in the Gray Scott model, we assume that he activator diffuses more slowly than does the inhibitor, so that s2 « D. The G M system exhibits surprisingly rich dynamics for various parameter ranges. Large am- plitude spike solutions have been studied intensively using numerical methods since the 1970's (cf. [26], [54], [33] and references therein), but only relatively recently from an analytical view- point. In this chapter we study asymptotically the dynamics of a one-spike solution to the G M system with r = 0 in the limit e —* 0. A one-spike solution has the form shown in Figure 3.1. 99 Z o o m o f H H A Figure 3.1: A spike for the Gierer-Meinhardt system (3.1) with r = 0 in a square domain with (p,q,r,s) = (2,1,2,0) (with A, H rescaled so that both are 0(1) as e —> 0). Here, e = 0.01, D = 5. Note that the inhibitor H does not change very much compared to A at the center of the spike. Before describing our specific results for (3.1), we survey some previous results on spike solutions to the G M system in a two-dimensional domain. When r = 0 and D is infinite, (3.1) reduces to the well-known shadow system involving a non-local scalar partial differential equation for the activator concentration A. The behavior of spike solutions to this shadow problem is now well understood (see [37], [10], [78]). As e —> 0, the equilibrium location of the spike for a one-spike solution is at the center of the largest ball that can be inserted into the domain (cf. [73], [81]). This solution is metastable in the sense that a single spike located in the domain moves exponentially slowly towards the boundary of the domain (cf. [37]). For the equilibrium shadow problem solutions with multiple spikes are possible. The locations of these spikes were found in [5], [27] and [49] to be related to a ball-packing problem. Equilibrium solutions for the shadow problem with two or more spikes are unstable on an 0(1) time scale. In the regime where D is at least logarithmically large as s —> 0, but not exponentially large, i.e. - l n e < D < 0 ( e 2 e 2 d / £ ) , where d is the distance of the spike center from the boundary, the stability of an equilibrium n- 100 spike pattern was analyzed rigorously in [83]: It was found that for e <C 1 there exists threshold values Di < D2 < ... such that an n-spike equilibrium solution is stable if and only if D < Dn. For the case of a one-spike solution, a differential equation for the dynamics of the center of the spike was derived in [14] and [76]. When D is very small, so that D = 0(e2), the motion of two spikes in M 2 was analyzed in [24]. In this case, both the activator and the inhibitor concentrations are localized near the core of the spikes, with A and H decaying exponentially away from the spike cores. It was found in [24] that the interaction between the spikes is exponentially weak and that the spikes move away from each other with a speed that is exponentially decreasing with the distance between the spike centers. A n explicit differential equation for the distance between the spike centers was derived. In this chapter, in addition to e2 <C D, we will also always assume: D <^0(e2e2dle). (3.3) where d is the distance of the spike from the boundary. The case of exponentially large D will be discussed in the following chapter. The results of this and the following chapter were previously reported in [46] and [47], respectively. We start by deriving a unified dynamical law that determines the motion of a single spike inside a two-dimensional bounded domain for any D with D 3> 0{E2). The previous results for large D found in [14] and [76], as well as results for small D, are then obtained as limiting cases. The motion of the spike is found to depend critically on various Green's functions and their gradients. The equation of motion for a spike for (3.1) when D = O( l ) and r = 0 differs significantly from the case when D 3> — In e, since for D » — In e only the gradient of the regular part of a modified Green's function for the Laplacian is involved (cf. [76]). However, when D = 0(1), we find that the differential equation for the spike motion involves both the regular part of a certain reduced-wave Green's function and its gradient. This complication results in part because-of the presence of the two different scales, e and — ^ 7 , that arise-due to the logarithmic point-source behavior of the two-dimensional Green's function. The presence of these two scales 101 Figure 3.2: A dumbell-shaped domain and its unique spike equilibrium location in the case makes the asymptotic analysis of the spike motion rather delicate. The second goal of this chapter is to examine how both the shape of the domain and the inhibitor diffusivity constant D determine the possible equilibrium locations for a one-spike solution. We find that for D small, the stable equilibrium spike locations tend to the centers of the disks of largest radii that can fit within the domain. Hence, for D small, there are two stable equilibrium locations for a dumbell-shaped domain. In contrast, we find that for a certain dumbell-shaped domain, there is only one possible equilibrium location when D is sufficiently large. Such a domain is shown in Figure 3.2. To obtain this latter result, we use complex analysis to derive an exact expression for the gradient of the modified Green's function for the Laplacian. While this result is obtained for a very specific dumbell-shaped domain, we conjecture that it is true more generally. More specifically, we conjecture that when D is sufficiently large there is only one possible equilibrium spike location for any connected domain. This conjecture is further supported through numerical experiments. The outline of this chapter is as follows. In §3.1 we introduce an appropriate scaling of (3.1), and we derive the equation of motion for a single spike, which is valid for any D satisfying D 3> 0(e2). In §3.2.1 and §3.2.2, we then derive limiting results of this evolution for the special cases where D <C 1 and D 3> — ln e, respectively. The exact solution for the modified Green's function of the Laplacian on a domain that is an analytic mapping of the unit disk is derived 102 in §3.3. This result is then applied in §3.3.1 to a specific dumbell-shaped domain. In §3.3.1, a conjecture regarding the uniqueness of the equilibrium spike location for large D is proposed. Numerical evidence supporting this conjecture is given in §3.4. In §3.4 we also compare our asymptotic results for the spike motion with corresponding full numerical results. 3.1 Dynamics Of A One-Spike Solution In this section we study the dynamics of a one-spike solution to (3.1) when r = 0. We assume that the spike is centered at some point x = XQ € O. The goal is to derive a differential equation for the dynamics of xo(t) for any D with D 3> 0(e2). We begin by introducing a rescaled version of (3.1) as was done in [84]. This scaling ensures that the rescaled inhibitor field / i i s 0 ( l ) a s £ — > 0 a t x = x o £ f i . To find such a scaling, we let A(x) = £a(x) and H = ^T~h(x), for some constant £ to be found. Wi th this change of variables, and setting r = 0 in (3.1b), (3.1) becomes x e O , t > 0, (3.4a) x e , t>0, (3.4b) p)(l + s). (3.4c) as e ^ O . (3.5) Since D 3> 0(e2), a spike core of extent 0(e) will be formed near a; = XQ. In the core, we define a new inner variable y — e~1(x — XQ). Outside of the spike core, where |y| —> co, the linear terms in (3.4a) dominate, and a decays exponentially as . a - Cell2\x - x 0 r 1 / 2 e - l " - x ° l / e , (3.6) for e~l\x — XQ\ —t oo. In the core of the spike, we assume that h changes more slowly as e —> 0 than does a. This arises from the assumption that D ^> 0(e2). In other words, for e —-> 0, we 103 where 7 is defined by 2 A A P at = e A a — a + — , hi T Q = DAh-h + Cv-, hs 7 = r + -(1 q The parameter £ will be chosen so that h(xQ) = 1 + o ( l ) , assume that to a leading order approximation a(x0 + ey)r a(x0 + ey)r / ,r a(x0 + sy)p h{xQ + eyy ~ -H(^)s a ( X 0 + £ y ) ' h(x0 + ey)q ~ ^ + ^ ' Under this assumption, the equilibrium solution to (3.4a) in the limit e —• 0 is a(x) ~ 1x1 ( e _ 1 | x — xo|) , (3-8) for some XQ, where w(p) is the unique positive solution of w" + -w -w + wp = 0, p > 0 , (3.9a) t o ( 0 ) > 0 , w'(0) = 0, w ~ cp~l/2e-p, as p ^ oo. (3.9b) Here c is a positive constant. Let G(X,XQ) be the Green's function satisfying , A G - - ^ G = -5(x - x0), x G f t ; <9nG = 0, x e dtl. (3.10) Let be the regular part of G defined by R(x, XQ) = G(x, x-o) + — In \x — xo\. (3-H) Then, the solution to (3.4b) is h(x0) = J^G(x,x0)^^dx. . (3.12) Since the integrand in (3.12) is exponentially small except in an 0(e) region near x = XQ, we get from (3.7), (3.8), (3.11) and (3.12), that, as e -» 0, Kxo) ~ ^ j f a ( - ^ l n ( £ M ) + R)™r(M)dv = C£2D'£) L™r(M)dy + ' (3'13) Thus, to ensure that h(xo) = 1 + o(l) as e —> 0, we must choose £ as where 6 and f are defined as / • O O J b= wr(p)pdp, v = — - r - » e , as e —> 0. (3.15) 7o _ ln(-) 104 Substituting (3.14) into (3.4) we obtain the scaled system P ' at = e 2 A a - a + ^ - , x e Q, i > 0 , (3.16a) 0=-DAh-h+T^^-, t > 0 . (3.16b) be hs Next, we derive a differential equation for the motion of the center XQ of the spike. Our main result is the following: Proposition 3.1.1 Suppose that D 3> 0(e2). Then, the trajectory x = xo(t) of the center of a one-spike solution to (3.16) satisfies the differential equation d X 0 f A w q ^ £" VRo, ase^O, (3.17) dt V P - 1/ ln ( i ) + 27Ti2 0 where Ro and its gradient are defined by •Ro = R{x0,x0), S7Ro=VxR(x,xo)\x:=xo. (3.18) Here R is the regular part of the reduced wave Green's function defined by (3.10) and (3.11). We now derive this result using the method of matched asymptotic expansions. Assuming that a decays exponentially away from x = xo, we have that ar/hs decays exponentially away from x o . Thus, from (3.16b), we obtain that the outer solution for h satisfies D 2TT6 JR2 hs(x0 + ey) where B —> 1 as e —> 0. The solution to (3.19) is h ~ 2TTBVG(X, XQ) = Bu [- ln(e|y|) + 2 7 r i ? ( x 0 + ey, x0)} , (3.20) where y = e~l{x — XQ) and G satisfies (3.10). The local behavior of the outer solution near the core of the spike is h~ B + 2irvBRa-vB\i\\y\ + 2-KevBVR0-y + O(e'2\y\2v), as x -* x0 . (3.21) The difficulty in matching an inner solution to the local behavior of the outer solution given in (3.21) is that there are two scales, u and e, to consider. To allow for these two scales, we must 105 expand the inner solution in a generalized asymptotic expansion of the form a = a0(\y\\v) + evai(y\v) + • • • , h = h0(\y\; u) + £vhx(y\v) + • • • , (3.22) where y = e~l [x - X0(T)] , r = e2ut. (3.23) Generalized asymptotic expansions of the form (3.22) have been used in [70] and [75] to treat related singularly perturbed problems involving the two scales u and e. Substituting (3.22) and (3.23) into (3.16), and collecting powers of e, we obtain P r A a 0 - a 0 + ^ | = 0, Ah0 + ^ = 0, | y | > 0 , (3.24) hi bhsQ and ( 3 - 2 5 b ) Here the prime on ao indicates differentiation with respect to |y|. The matching condition is that Oi —> 0 exponentially as |y| —> oo and that h satisfies (3.21) as |y| —> co. We first study the problem (3.24) for the radiallly symmetric solution ao and ho- Since the outer inhibitor field is to satisfy h(xo) = 1 + o(l) as e —> 0, we expand the solution to (3.24) as / i 0 = l + ^ o i ( | y | ) + 0 ( ^ 2 ) , a 0 = ™(|y|) + ^aoi(|y|) + C V ) . (3.26) Here ui is defined in (3.9). Substituting (3.26) into (3.24), we obtain for |y| > 0 that Aao i — aoi +pwp~1aoi = qwphoi, (3.27a) Ah01 + lwr = 0. (3.27b) b The matching process then proceeds as in [70] (see also [75]). Since v S> e, we treat v as a constant of order one in the local behavior of the outer solution given in (3.21). We now match 106 the constant term of the inner solution ho to the constant term of the local behavior of the outer solution (3.21). This yields 1 = B + v2ivBRo, so that B = i n~~7>— " ( 3 - 2 8 ) 1 + 2-KRQU V ; Substituting this value of B back into (3.21), we then obtain the revised matching condition h ~ 1 - 1 , ^ _ ln \y\ + 2UEVBVRO • y + • • • , as y -> oo , ,(3-29) where B is given in (3.28). Expanding (3.29) in a Taylor series in v, and comparing with the expansion of ho in (3.26), we conclude that hoi must satisfy (3.27b) subject to the far-field asymptotic behavior hoi — — In \y\ + o ( l ) , as \y\—> oo. (3.30) Recalling the definition of 6 in (3.15), it easily follows that there is a unique solution to (3.27b) with asymptotic behavior (3.30). Solving for hoi, and then substituting into (3.27a), we can then in principle determine ani- Higher order terms in the logarithmic expansion of an and ho can be obtained in the same way. We now study the problem (3.25) for ai and hi. From the matching condition (3.29) it follows that we must have hi = 2TTBVRO • y + o(l) as \y\ —> oo. Thus, we introduce hi by hi = 2vrBV.Ro -y + hi, (3-31) where hi —+ 0 as \y\ —* oo. Substituting (3.31) into (3.25), we can write the resulting system in matrix form as L<p + Mcp = f, M ^ ( m U ^ | , 0=( y | , / = | " | , (3.32) m 2 i m 2 2 where L is the Laplacian operator Lcf> = Acp, and p—l P < _ qap mn = - l + ^ V , m12 = - - ^ r , (3.33a) *""a0 usaQ ftm, ™2i = , , „ , m 2 2 = -771TT' (3.33b) fi = 27rgJBVi?0 • y ^ - a o ^ p , h = ZKVSBVB* • y ^ • (3.33c) 107 The solution to (3.32) must satisfy <j> —> 0 as \y\ —» oo. To derive the differential equation for xo(t) we impose a solvability condition on (3.32). Let tp be any solution to the homogeneous adjoint problem associated with (3.32). Thus, tp satisfies, LtP + MttP = 0, (3.34) with tp —> 0 as \y\ —> oo, where M1 indicates the transpose of A4. Multiplying (3.32) by tp1, we integrate by parts over R 2 to obtain f (tfLd* + tfM<f>) dy= f qbl (Lib + Mty) dy = f tp1 f dy. (3.35) J IR2 J R 2 J K 2 Since V satisfies the homogeneous adjoint problem, we conclude from (3.34) and (3.35) that (3.32) must satisfy the solvability condition tptfdy = 0. (3.36) R2 We now obtain a more convenient form for this solvability condition. Setting tp = (ipi,^)1, and using (3.33a) and (3.33b), we write the adjoint problem (3.34) as * * + ( - 1 + £ f ) * + ! : $r* - 0 ' ( 3 3 7 a ) A ^ - ^ , - ^ f c = 0, (3.37b) where tpj —» 0 as \y\ —* oo for j = 1,2. Using (3.33c), the solvability condition (3.36) can be written as 4 ' / fL{a'0iP1dy = 2nBVRo- [ y ( + ^fr) dy . (3.38) Equation (3.38) is simplified further by using (3.37b) to replace the right-hand side of (3.38). This yields, x'o • [ h a'o^ dV = Vf lo • / • y A ^ 2 dy , (3-39) 7R2 |y| 7R2 where B is defined in (3.28). Equation (3.39) is an ordinary differential equation for the motion of the center of the spike. 108 We note that the derivation of (3.39) has not used any expansion of an or ho in powers of the logarithmic gauge function v. In principle, to determine an explicit form for the O D E (3.39) for 2n(£) , which contains all the logarithmic terms, we must solve (3.24) for an and ho and then compute non-trivial solutions to the adjoint problem (3.37). This is a difficult task. Instead, we will only calculate the leading order term in an infinite logarithmic expansion of and ipi • This requires only the leading order term in the infinite logarithmic expansion of ao and ho given in (3.26). Therefore, substituting (3.26) and V i = tpw + + 0(v2), V2 = 2̂0 + W>21 + 0(u2), (3.40) into (3.37), we obtain the leading order adjoint problem A ^ 1 0 + (-1 + pwP-1) VJ10 = 0, ' (3.41a) A ^ 2 0 - qwpyj10 = 0. (3.41b) There are two linearly independent solutions to (3.41a). They are Vio = dyjw , j = l,2. Substituting (3.42) into (3.41b), we obtain A V 2 o = -^ I K^(M)] 'g, 3 = 1,2. The solution to (3.43) is P+ i u \y\ (3.42) (3.43) (3.44) where p = \y\. Substituting ao ~ w, (3.42), and (3.43), into the solvability condition (3.39), we obtain x. o- , ^ w ' d y j w d y = ^ V R 0 - j y K+ 1(|y|)]'pr^, i = l , 2 . (3.45) /R2 \y\ P The integrals in (3.45) are evaluated using ytyj R2 \y\2 ' ~\2 f°° r ' i 2 w (\y\) dy = TT5i:j / p w (p) dp, J Jo L J oo (3.46a) / m dy = TfSij I p2 [wp+1(p)\ dp = -2TT<% / p[w(p)}p+1 dp, (3.46b) / R 2 \y\ Jo , Jo 109 where 5{j is the Kronecker symbol. Substituting (3.46) into (3.45), we obtain p + 1 V Jo iw (P)\ PdP ) In Appendix B of [76], equation (3.9) was used to calculate the ratio fo°°Mp)}P+1pdp _P+1 f0°°[w'(ptfpdP P - r Hence (3.47) reduces to (3.48) X'0(T) = - ^ V R 0 . (3.49) Substituting (3.28) for B into (3.49), and recalling the definition of v given in (3.15), we recover the main result (3.17) for xo(t). There are two important remarks. Firstly, from (3.49) it follows that the center of the spike moves towards the location of a local minimum of Ro- This minimum depends only on D and not on £. In the following sections we will explore how this location depends on D: Secondly, as seen from the analysis above, since we have only used the leading order term in the logarithmic expansion of the homogeneous adjoint eigenfunction, the error in (3.17) is of order 0(u). This error, however, is still proportional to Vi?o- In fact, the two integrals in the solvability condition (3.39) are independent of XQ and the shape of the domain. Thus, even if we had retained higher order terms in the logarithmic expansion of the adjoint eigenfunction, we would still conclude that the equilibrium locations of the spike are at local minima of Vi?o, and the spike would follow the same path in the domain as that described by (3.17). The higher order terms in the logarithmic expansion of ao, ho and the adjoint eigenfunction, only change the time-scale of the motion. However, at first glance, an error proportional to 0(u) in the time-scale of the asymptotic dynamics seems rather large. This is not as significant a concern as it may appear, as from the numerical experiments performed in §3.4 we show that it is the dependence of B on v as given in (3.28) that allows for a close agreement between the asymptotic and full numerical results for the spike motion. To derive Proposition 3.1.1, we had to expand the ao, ho in v as in (3.26). But this is only valid when v is small, i.e. e is exponentially small. To derive more accurate dynamics for a more realistic case of e polynomially small, we need to work directly with adjoint equations (3.37). 110 Let r = \y\. Direct computation yields, for any function / ( r ) , A f e / M ) = % ( / » + ^ / ' « ) . Letting we obtain M*) = M^) = ^9(r), (3.50a) and —a'0yjidy = % T T / ra 0 (r) / ( r )oY ( 3 . 5 1 ) y Jo Therefore we obtain: 4 ^ r a , ( r ) / ( r ) , / (3.53) The integral on the right hand side is to be evaluated numerically. The functions ao, ho, f, g are solved numerically using a boundary value solver co l sys [3]. We approximate the infinite line by [0, L] where L was taken to be 1 5 . Note that / and g are determined only up to an arbitrary scaling. For / , g we use the following boundary conditions: / ( 0 ) = 0 , g'(0) = 1, f'(L) = 0 , g'(L) = 0 . These conditions assure decay at infinity and the solution being odd. The remaining conditions are: a'0(L) = 0 , h'0(Q) = a'0(0) = 0 , and h(0) = 1. Next we consider an example. We take tt = [0, l ] 2 , e = 0 . 0 1 , (p, q, r, s) = ( 2 , 1 , 2 , 0 ) . Taking an initial spike to be at XQ = ( 0 . 4 , 0 ) we obtain numerically that Ro = 0 . 7 5 9 8 and V J R O = 1 1 1 (—0.09137,0) (see Section 3.2.1.) From a full numerical simulation of the system, we obtain ' f f i f f 1 = 7.057. On the other hand, using (3.28) we obtain B = 0.491 and = 6.170, with an error of about 10%. Using co lsys obtain numerically, ,oo ° = -2.566 Jo raof which yields from (3.53), ^ y ^ ^ = 7.92. This method also give an error of about 10%. However we expect that the latter method is more accurate for smaller values of e. 3.2 L i m i t i n g Cases O f T h e Dynamics In this section we consider two limiting cases of result (3.17) for the motion of a spike. In §3.2.1 we consider the case where £ 2 « D « 1 and in §3.2.2 we consider the case D 3> 1. 3.2.1 Dynamics For Small D In this section we assume that £ 2 C D <S 1. The inequality £ 2 < D was crucial to the derivation of (3.17) in §2. When J ) < 1 we can treat D as a small parameter and obtain limiting results from (3.17). We begin by introducing R defined by R{x,xQ) = G{x,xQ)-V(x), (3.54) where G satisfies (3.10), and V is the free-space Green's function in R 2 satisfying A V - X2V = -5(x - s 0 ) , ' A E E - ^ . (3.55) The solution to (3.55) is V{X) = ^-K0(\\X-XQ\). (3.56a) The asymptotic behavior of K~o(r) for r -C 1 is K0(r) ~ - l n r + l n 2 - 7 + 0 ( r 2 l n r ) , as r -> 0. (3.56b) Here 7 is Euler's constant. In terms of R, the regular part RQ defined in (3.18) is Ro = R(x0,x0) - ^ - ( l n A - l n 2 + 7 ) , WRo = VR(x,x0)\x=xo . (3.57) 112 To obtain some insight into the dynamics when D is small, we first consider the case where fl = [0, l ] 2 is a unit square. Then, us for G for any value of D. This yields ing the method of images, we can solve (3.10) explicitly ( oo oo \ 2 V bm(M*))] - V(x), (3.58a) n=-oom=-oo / where . (xi — n, x 2 ) if n is even I (x i , x 2 — m) if m is even hn(x) = \ , vm(x) = < (n + 1 — x i , x 2 ) if n is odd I ( x i , m + l — x 2 ) if m is odd (3.58b) For D small, such that A|x — xo| 3> 1, the function V decays exponentially as V(x) ~ i ^ = [ A | x - x 0 | ] - ^ e - A l x - x ° l , for A | x - x 0 | » 1. ' (3.59) 2 \ /27r Now suppose that the spike is located at xo = (^,0 with O (j) -C £ < \ — O (\). Then, for D C 1, we need only retain the two terms (n,m) = (0,0) and (n,m) = (0,-1) in the series (3.58a). The other terms are exponentially small at the point xo in comparison with these terms. Thus, for A —> 00, we obtain from (3.58a) that R(x,x0) ~ [A|x - x 0 | ] , x = ( x i , - x 2 ) , x = ( x ! , x 2 ) . (3.60a) Now substituting (3.60a) into (3.57), and using the large argument expansion (3.59), we obtain Ro ~ 7 ^ f - 2 A ? + I (m 2 - 7 - in A) , 2VR0 ~ j , (3.60b) where j is a unit vector in the positive x 2 direction. Substituting (3.60b) into (3.17), we obtain an evolution equation for £ d£ q ( e2V^X \ e- 2 A^ dt p - 1 I l n 2 - 7 - l n [ e A ] / ' (3.61) We now make a few remarks. The O D E (3.61) breaks down when eX = 0(1). This occurs when D = 0(e2). Thus, we require that £ « 1 and A > 1, but eA < 1. In this limit, (3.61) shows that £ is increasing exponentially slowly without bound as t increases. The O D E , however, was 113 logio t Figure 3.3: Movement of the center (0.5, £(£)) of a single spike of (3.16) within a unit box [0, l ] 2 versus log 1 0 £ , with e = D = 0.01. The solid curve is the numerical solution to (3.61) with £(0) = 0.2. The broken curve is the approximation (3.63). derived under the assumption that 0 Q ) <C£ < \ — O (j). When £ is near the value £ = 1/2, the O D E must be rederived by retaining an additional image point in the infinite sum in (3.58a) corresponding to (n,m) = (0,1). The effect of this additional term is to ensure that £ —> 1/2 as t —> oo. This implies that the spike tends to the center of the square as t —* oo. Consider (3.61) with the initial condition £(0) = £o- To determine the time T for which £(r) = £i, where O (\) <C £o < £i < \ - O (\), we integrate (3.61) to obtain Thus, when D < 1 , the motion of the spike is metastable. The spike moves exponentially slowly with time (see Fig. 3.3) as it approaches the center of the square. Indeed, this behavior is not specific to a square domain as we will now demonstrate. (3.62) Evaluating the integral asymptotically for A > 1 we get A > 1. (3.63) More generally, consider any domain ft with smooth boundary. Let R(x, x0) be defined as in (3.54). Then, R satisfies AR — X2R = 0 x 6 ft; dnR = -dnV, x e 5 f t , (3.64) 114 where V(x) is given in terms of XQ by (3.56a). To obtain a representation formula for R, we apply Green's theorem to R and V. This yields, Ro = R(xo, XQ) = / V(x )dnR(x ,XQ) — R(x ,xo)dnV(x) dx . (3.65) Jan L -1 The only term in the integrand of (3.65) that we still need to calculate is R(x ,xo) for x G dQ. We now calculate this term for A > 1 using a boundary layer analysis on (3.64). Since A > 1, the solution to (3.64) has a boundary layer of width O ( A - 1 ) near dQ. Thus, it suffices to estimate R inside the boundary layer. Let ri = A|x' — x\ where x is the point on dQ closest to x (one can always find such an x assuming that x is within the boundary layer and A is sufficiently large). Let £ represent the other coordinate orthogonal to rj. Then, using this coordinate change in (3.64), we have to leading order that #^-5 = 0, 77 > 0 ; X R ^ ~ dnV(x). (3.66) Since XQ is assumed to be strictly in the interior of Q, we can estimate V on dQ using the. far field behavior (3.59). This yields, for A > 1, that dnV(x)~-XV(x')(f',n), r = * ~ *° , (3.67) where h is the unit outward normal to dQ at x , and the angle brackets denote the scalar dot product. The solution to (3.66) that is bounded as 77 —> +00 is proportional to e~v. Therefore, £~ -\-ldnV(x')e-r'. (3.68) Using (3.67), and evaluating (3.68) on dQ where 77 = 0, we obtain the following key results for A » 1: ' R{x\x0) ~ V{x)(r , n ) , x G dQ, (3.69a) dnR(x',xo)~\V(x)(r,n), x G dQ. (3.69b) Next, we substitute (3.69) and (3.67) into (3.65). This yields, for A > 1, that R0 = R(x0, x0) - A f \v(x)]2 ((r', n ) 2 + <f, n)) dx . (3.70) JdQ L J 115 We now evaluate this integral asymptotically for A ^> 1. To do so we use Laplace's formula (see [86]), Here r m = dist(d£7, xo), ftm > 0 is the curvature of dfi at x m , and the sum is taken over all x m G 9fi that are closest to xo- Comparing (3.70) with (3.71), we get F(r') = ±-((r',n)2+(r',n)) . (3.72) At the points x m 6 dfi closest to xo, we have that r = rm and r = h. This yields, F(rm) = 1/47T. Therefore, for A » l , the estimate (3.71) for (3.70) becomes Ro === e-2Xr™ y f l - ^ ) 2 . (3.73) Finally, to calculate Vi?o needed in (3.17), we use (3.57) and the reciprocity relation R(x, xo) = R(XQ, x) to get 1 d 1 d V i ? 0 = VR(x,x0)\x=X0 = - — R ( x 0 , x 0 ) = ^-T-RG- (3.74) Zaxo 2axo Differentiating (3.73), and substituting into (3.74), we obtain where fm = ( x m — x o ) / | x m — xo|. Substituting (3.75) and (3.57) into (3.17), we obtain the following proposition: Propos i t i on 3.2.1 For e 2 <C D <C 1 and e <C 1, the trajectory of the center of a one-spike solution to (3.16) satisfies the differential equation d X 0 ^ X q { £ " ^ 1 e - ^ - W 1 - - ) ^ . (3.76) dt p - l V l n 2 - 7 - l n [ e A ] 7 ^fF, Here A = D~z, fm is defined following (3.75), and the other symbols are defined in the sentence following (3.71). The formula (3.76) agrees with (3.61) when fl is a unit box. Moreover, we have the following result: 116 P r o p o s i t i o n 3.2.2 Let r(x) = dist(5ft,x). Suppose that xo is a local minimum of /(xo) = R(xo, xo) as A —> co. Then, for A —> oo, xo is a local maximum of r{x). Proof . The proof is by contradiction. Suppose that xo is not a local maximum of r(x). Since r is continuous, we can find x\ with | x i — XQ\ > 0 arbitrary small, with r(x\) — r(xo) > 0. However, (3.73) yields • g f r i ^ i ) _ C e - 2 A [ r ( x 1 ) - r ( x 0 ) ] ^ (377) R(X0,X0) where C = G(xo ,x i ) is independent of A. Hence, for A sufficiently large, ~^ X 1 ' X 1 ^ < 1. This R(xo,xo) implies that R(xi,x\) < R(XQ,XO) for A large enough. Hence, xo is not a local minimizer of R as A —> oo. Using (3.57) to relate R to R completes the proof. • It follows that for D -C 1 and for convex domains, the center of the spike moves towards a point within the domain located at the center of the largest disk that can be inserted into the domain. 3.2.2 Dynamics For Large D The dynamics for the limiting case where D ^> 1 is significantly different from the previous analysis where D -C 1. When D is large, we may expand G defined in (3.10) as G = DG0 + Gm + ^G2 + ••• . (3.78) Substituting (3.78) into (3.10), and collecting powers of D, we obtain A G 0 = 0, x G ft; d n G 0 = 0, x G dQ, (3.79a) A G m = Go - 6{x - x 0 ) , xeQ; dnGm = 0, xedQ. (3.79b) From (3.79a) we conclude that Go is a constant. The solvability condition for (3.79b) then yields 117 where volf i is the area of Cl. The solvability condition for the problem for G2 also yields that Jn Gm dx = 0. Hence, Gm is the modified Green's function for Cl satisfying AGm = — j — - 5(x -x0), i € f l ; dnGm = 0, x&dCl; f Gm dx = 0. (3.81) vols 2 7n Let Rm be the regular part of Gm defined by Rm(x, XQ) = ~ In \x - x0\ + Gm(x, x0). (3.82) Combining (3.11), (3.78), (3.80), and (3.82), we conclude that for D > 1 R(x,x0)~-^ + Rm(x,xo) + O(l/D) . (3.83) Substituting (3.83) into (3.17), we obtain the following proposition: P r o p o s i t i o n 3.2.3 / / D S> 1 and e <C 1, the trajectory of a one-spike solution of (3.16) satisfies ^ = L _ )vRmo, (3.84a) dt p-1 \-\ne + 2 ^ + 27rRmoJ °' ^ where Rmo and its gradient are defined by Rm0 = Rm{xo,x0), VRmo = V x i ? m ( x , x 0 ) | x = x 0 • (3.84b) As a corollary, we obtain the following proposition, which was originally derived in [14] and [76]: P r o p o s i t i o n 3.2.4 (Ward et al, [76], Proposition 3.2) Let e < 1 and assume that D > - Ine. Then, the trajectory of a one-spike solution of (3.16) satisfies i X 0 - ^ 5 V ° l f i ^ V i C o , (3.85) dt . \ p - l ) D where Rmo and VRmo ore defined in (3.84b). 118 Several general observations can be made by comparing (3.17), (3.76), and (3.85). In all three cases, the activator diffusivity e only controls the timescale of the motion. The precise trajectory traced by xo as t increases depends only on D and on the shape of the domain inherited through the terms Vi?o and V i ? m o - For the case of small D, where e1 -C D <C 1, the motion is exponentially slow, or metastable, and D controls the dynamics. In contrast, when D = 0(1), 2 the speed of the spike is of the order _ £ l n e , with a complicated dependence on D through Ro. 1 Finally for D ^> — lne, the speed is controlled by both e and D and is of order ^ . In the limit D —> oo and r = 0, the system (3.1) is approximated by the so-called shadow system (see [37], [10]). In this case the motion of the spike is again metastable. However, for the shadow system a one-spike interior equilibrium solution is unstable and the spike moves towards the closest point on the boundary of the domain. This behavior is in direct contrast to what we have found for small D, whereby by Propositions 3.2.1 and 3.2.2 the spike moves exponentially slowly towards a point that is the furthest away from the boundary. This suggests that as D is increased, the number of possible equilibria for XQ may decrease. In §3.3 we will show that for a certain dumbell-shaped domain, there is only one possible stable equilibrium location for D sufficiently large, whereas there are two stable equilibrium locations when D is sufficiently small. The analysis of this chapter breaks down in the near-shadow system when D is exponentially large, i.e. when (3.3) fails. In such a case, there is an exponentially weak boundary effect that must be taken into account, since it becomes of the same order as the effect due to V i ? m . The near-shadow system will be considered in detail in the next chapter. 3.3 E x a c t Ca l cu l a t i on O f T h e M o d i f i e d Green 's F u n c t i o n In this section we will use complex analysis to derive an exact formula for VRmo defined in (3.84b) for domains of the form « = / ( £ ) , (3.86) where B is the unit circle, and / is a rather general class of analytic functions. We will then use this formula to further explore the dynamics of a spike for the G M system on a certain 119 dumbell-shaped domain. Our main result here is the following: Theorem 3.3.1 Let f(z) be a complex mapping of the unit disk B satisfying the following conditions: (i) f is analytic and is invertible on B. Here B is B together with its boundary dB. (ii) f has only simple poles at the points z\,z<i, and f is bounded at infinity. (iii) f = g/h where both g and h are analytic on the entire complex plane, with g(z{) ^ 0. (iv) f(z) = f(z). On the image domain $7 = f(B), let Gm and Rm be the modified Green's function and its regular part, as defined in (3.81) and (3.82), respectively. Let Rmo and VRmo be the value of Rm and its gradient evaluated at XQ, as defined in (3.84b). Then, we have V t f m 0 = ^ , (3.87) / Oo) where ZQ G B satisfies XQ = /Oo); and Vs(zo) is given by v? r \ 1 (  z° , /"(*o) \ VsOo) =7T- r, r-To + 2rr V 1 - ko|2 2/'(z 0) 1 A ^ 0 O i)/'(i) / 1 /'Oo) (/oo) - / ( ! ) ) + E 9^hr + T - M (3-88) V zQ J ^ zjh (zj) \ZJ - z0 1-ZJZQJ V ' In the equation above, and for the rest of this section, we will treat vectors v = Oi,t>2) as complex numbers v\ + iv2- Thus vw is assumed to be complex multiplication. The dot product will be denoted by (v,w) = \(vw + vw). Proof. Given x,xo € Q, choose z, ZQ such that x = f(z),xo = /Oo). We will use n and N to denote the normal to dB at a point z and the normal to dQ at x = f(.z), respectively. Since / is analytic on B we obtain ft nf'jz) zf'(z) dz N = WW\=WW\' ^ = Tz' ( 3 - 8 9 ) 120 where da is the length element on dB. We now define S(x) by 1 1 S(x) = Gm(x,x0) + — ln |x - x 0 | - - — — |x - r E 0 | 2 .' (3.90) In 4vol&2 Substituting (3.90) into (3.81), we find that S satisfies A S = 0, in ft, (VS,N) = ( x - x 0 , N ) ( ^ ^ - J ^ m ) , on (3.91a) with S dx = — / ln lx —xnl r - r / Ire — rcnl2 ĉ c • (3.91b) Combining (3.90) and (3.82), we relate Vi?™ to V S as VRm(x,x0)\x=xo = VS(x0). (3.92) The problem (3.91a) determines S up to an additive constant. This constant is determined by (3.91b). However, the precise value of this additive constant does not influence S/Rmo, since this term depends only on the gradient of S. Hence, without loss of generality, in the derivation below we only calculate S up to an additive constant. Let s(z) = S(f(z)). Since / ,is analytic and S is harmonic, s satisfies Laplaces's equation.. Using (3.89), and the fact that / is analytic, we get (Vs,n) = (VS,N)\f' (z)\. Hence, (3.91a) transforms to A s = 0, in B, (Vs, ft) = X{z, ZQ) = (X- X 0 , zf (z)) (2v]xlxop ~ 2̂ oTQ j » on dB. On the unit ball fl = B, let gm(z,£) be the solution to the modified Green's function problem (3.81), with singular point at z = £. It [76] it was shown that 9m(z, 0 = ^ - l n I* - £| - In |z - l) + C(0 , (3.94) where C is a constant depending on £. Notice that if |z| = 1 then \z — J^I\2 = • Hence, V*9m(z, 0\zeaB = + > (3-95) 121 where C\ is another constant. Next, we use Green's identity to represent the solution to (3.93) as the boundary integral *(£) = / 9m(z,0x(z,zQ)da(z) + C2, X ( z , z0) = (Vs.n), (3.96) JdB where C2 is a constant. Since s(z) = S(f(z)), we get from (3.92) that Vs(z) I VRmo = V i ? m ( x , x 0 ) | x = ; r (3.97) Differentiating (3.96) with respect to £ and using (3.95), we evaluate the resulting expression at £ = ZQ to get Vs(z 0 ) = / \/igm(z,z0)x(z,z0)da = - / — x O , z0) da + Ci f x(z,z0)da. (3.98) JdB  7 7 iaB \ z ~ z0\ JdB From (3.93) it follows that JgB x(z, z0) da(z) = 0. Then, using (3.93) for x(z,z0) and (3.89) for da(z), (3.98) becomes Vs(z0) = Jjx-x0,.zf(z)) ( 2 ^ ^ 0 | 2 - ^ ) ^ j J ^ l J = h JdB Ga - -o)^7>)+^. / '0) ) ( 2 7 r | x^ X o | 2 - 7 7 ^ ^ 4vr 2i 7a B - 1 - ZZQ ^ 47r 2i 7 a s x - x0l - zz0 —~ f (x-xo)zf'(z) dz , ^ / x-XQzf(z)-— dz . Amvoinjgg J J y j l - z z Q Airi vol n J9B J K J l - z z 0 (3.99a) This equation is written concisely as Vs(zo) = J1 + J2 + J3 + M, (3.99b) where the Jk are, consecutively, the four integrals in the last equality in (3.99a). We now calculate each of these terms. We first calculate J2. Using the residue theorem, the relations x = f(z) and xo = /(^o), and the invertibility of / , we readily calculate that 4nzi JdB x - x 0 1 - zz0 2rr [1 - | z 0 | J 122 To calculate J i we use an identity. For any function H(z), it is easy to show that 2m This determines J\ as - / H{z)dz = ^ - f H(z)\dz. (3.101) JdB 2mJ9B z2 v / 1 f fzf(z)\ 1 , f (z0) J l = mr- i — — D Z = A f > , - \ • 3 - 1 0 2 <±-xh JdB \x - xQ j l - ZZQ 4TT/ (z0) To show this, we use zz = 1 for z G <9i?, and (3.101) and (3.102), to get J l = l W [( f { U ] ^U^ lV/ 7 ~ ^ T T ^ - ( 3 - 1 0 3 ) 4wJdB \{x - x0){z - z0) J z2 ATT\ JdB {x - x0)(z - z0) Using x = f(z) and x 0 = /(-zo), we write (3.103) as 4TH* 79B (z - za)2 f(z) - f(z0) The function cf>{z) is analytic in B and 4> (ZQ) = f" (zo)/ 2f'(ZQ) • Thus, using the residue theorem, and property (iv) of Theorem 4.1, we get J ' = ^ = S 7 W - ( 3 I 0 5 ) This completes the derivation of (3.102). To calculate J3 and J4 , we need to evaluate vol ft given by vol ft = / dx= f -(x,N)dZ. (3.106) Jn Jan 2 Using dE = \f (z)\da, x = / (z ) , zz = 1 on 9 5 , and (3.89) for TV and da, we get volft = -W J(z)f'(z)dz + l I J{z)fMdz. (3.107) To evaluate (3.107), J 3 , and J 4 , we need another identity. Let F(z) be any function analytic inside and on the unit disk, and assume that F(z) = F(z). Then 'dB Y To show this result, we use (3.101) to write / as 1 2m -I z 2f(z)F(z)±dz=^- [ z2f{z)F{z)dz. (3.109) m JdB zA 2m J d B 123 Since zz = 1 and F(z) = F(z) = F (l/z). on dB, we get I = J. where J = mF(i/z) dz. (3.110) 27ri jaB Then, since F(l/z) is analytic in \z\ > 1 and f(z) is bounded at infinity by property (ii) of Theorem 4 . 1 , we can evaluate J by integrating the integrand of J over the boundary of the annulus 1 < \z\ < R and letting R —> oo. By properties (ii) and (iii) of Theorem 4 . 1 , / (z) = g(z)/h(z) has simple poles at z = Zj with \ZJ\ > 1. Using the residue theorem over the annulus, and letting R —> oo, we obtain • f z j h ( z j ) zj' (3.111) Substituting (3.111) into (3.110) and using f(z) = f(z), F(z) = F(z), and the fact that Zj is a pole of f(z) if and only if Zj is, we obtain the result (3.108) for I. Next, we use (3.101) to write vol ft in (3.107) as vol = rr ~ f f{z)f\z)dz+^-. f f(z)f'(z)dz 2m JdB 2 m JdB (3.112) Then, using (3 .108) , we can calculate the two integrals in (3.112) to get vol ft 7T E + h'(zj) z2h'(zj) 9{zj)f\i) (3.113) The last equality above follows from property (iv) of Theorem 4 . 1 , which implies that Zj is a pole of f(z) if and only if Zj is. Next, we evaluate J4 of (3 .99a) . We calculate that J 4 " " i ^ o m L f { z ) - f { Z 0 ) Z f ' { z ) l ^ o i Z = ^ ^ z^i^-zo) • ( 3 ' 1 1 4 ) To obtain this result, we use the fact that zf (z)/(l — ZZQ) is analytic in B to get dz 2 v o i n \2-KI JdB x 'I- zz0 The result (3.114) then follows by using the identity (3.108) in (3 .115) . (3.115) 124 Lastly, we calculate J3 of (3.99a). We find, 1 4 7 U V o l f t JgB •{f(z)-f(zQ)]zf'(z) 1 _ 1 ^ 9) 1 - Z Z 0 1 2 vol ft ^ / i ' ( Z i )zj (1 - ZJZQ) 2 vol ft To obtain this result, we use (3.101) to rewrite J3 as dz /Oo) - / ( - ) (3.116) J.3 = 1 — / f(z)-f(z0)F(z)dz F(z] (3.117) 2 Vol ft \ 27T2 JQB " J Z — ZQ Now we repeat the steps (3.109)-(3.111) used in the derivation of (3.108), except that here we must include the contribution from the simple pole of F(z) at z = ZQ, which lies inside B. Analogous to (3.110), we obtain 1 ( I f [f(z)-f(z0)]f'(l/z) X J3 2 vol ft \ 2vri JaB ^ ( Z - 1 - ZQ) dz (3.118) Outside the unit disk the integrand has simple poles at z = Zj and at z = 1/zo- Integrating over the annulus 1 < \z\ < R, using the residue theorem, and then letting R —> 00, we obtain (3.116). Finally, combining (3.99b) and (3.97), we obtain our result for the gradient of the modified Green's function 1 J L (3.119) VRmQ EE VRm(x,XQ)\x=xo = jj—j Jk . Substituting the results for Jk and volft given by (3.100), (3.102), (3.113), (3.114), and (3.116), into (3.119), we obtain our main result (3.87) and (3.88). ' • 3.3.1 Uniqueness O f T h e One-Spike Equ i l i b r i um Solu t ion For Large D Consider the following example from [30]: (3.120) Here a is real and a > 1. The resulting domain ft = f(B) for several values of a is shown in Fig. 3.4. Notice that ft —> B as a —> 00. One can also show that a s £ = a — l — > 0 + , f t 125 Figure 3.4: Left: The boundary of $7 = f(B), with / (z) as given in (3.120), for the values of a as shown. Right: The vector field VRMQ in the first quadrant of Q with a = 1.1. approaches the union of two circles centered at ( ± 1 , 0 ) , with radius ^, which are connected by a narrow channel of length 2e + 0(e 2 ) . From Theorem 3.3.1, we calculate Vs (z 0 ) z0 (ZQ + 3a 2) zQ Q?ZQ 2TT , I - N 2 ,4 _ i \ 2 zz - aH + zla2 + ZQ zl - a 2 (a 4 - l)2(lzol2 - l)(zp + a2z0)(z2 + a 2) (a* + l)(z2a2-l)(z2-a2)(z2-a2)2. (3.121) In the limit a —> oo, —> 73, x 0 —> zo ; and / '(0) —> 1. In this limit, we calculate from (3.87) and (3.121) that 1 / 9 _ I T J 2 \ (3.122) 1 /2- I X Q I 2 ' Vit!mo = — 7-^ x0 . 2TT V 1 FO This is precisely the formula for V i ? m o on the unit disk, which can be derived readily from (3.94) as was done in [76]. This provides an independent verification of a limiting case of Theorem 3.3.1. We have also verified the formula (3.121) by using the boundary element method to compute S7Rmo for Q as obtained by the mapping (3.120). The two solutions are graphically indistinguishable. Next, we calculate from (3.121) that V s ( z 0 ) | a ^ 1 + = Re(z0) 7r(l-|zo|2)(z2 + l ) (3.123) 126 In the limit a —> 1 +, ft becomes the union of two disks of radius 1/2 centered at ( ± ^ , 0). Thus, the unique root of Vs(zn)|a->1+ = 0 in ft is zo = 0. This root is easily verified to be a simple root. Hence, it follows from the implicit function theorem that Vs(zo) has a unique root for any E = a — 1 > 0 small enough. B y symmetry, this root must be at the origin. We summarize our result as follows: P r o p o s i t i o n 3.3.2 Consider a domain ft = f(B) with f given by (3.120) as shown in Fig. 3.4- Then for £ = a—I > 0 small enough, ft is approximately a union of two disks of radius ^ centered at ( ± ^ , 0 ) , connected by a narrow channel of size 2s + 0 (e 2 ) . Furthermore, VRmo given by (3.84b) has a unique root located at the origin. Thus, in this case, there is a unique equilibrium location for the single-spike solution of (3.16). We now show that there are no roots Vs(zo) along the real axis when a > 1, except the one at ZQ = 0. Thus, there are no equilibrium spike-layer locations in the lobes of the dumbell for any a > 1. In (3.121) we let ZQ — ZQ = £, where — 1 < £ < 1. After a tedious but straightforward calculation, we get V*(*o) = J^(0, ( 3- 1 2 4 a) 2 o V + 1) - (£ 2 + a 2 ) 2 , 1 f 2 (a 4 - l ) 2 ( a 2 + 1)(£ 2 + a 2 ) ( £ 2 - 1) ( a 4 - £ 4 ) ( l - £ 2 ) a 2 £ 2 - l (a 4 + l ) ( a 2 - £ 2 ) 3 (3.124b) The function \i is even. Thus, to establish our result, we need only show that /i(£) is of one sign on the interval 0 < £ < 1 for any a > 1. A simple calculation shows that the term in the square brackets in (3.124b) vanishes at £ = 1/a. In fact, £ = 1/a is a removable singularity of p,. It is also easy to show that //(0) > 0 for any a > 1, [i —> +00 as £ —> 1 _ , and / / (£) > 0 on 0 < £ < 1. Hence, for any a > 1, /i(£) > 0 on 0 < £ < 1. Thus, VS(ZQ) has a unique root at ZQ = 0. Consequently, there is only one equilibrium spike location, and it is at ZQ = 0. This leads us to propose the following conjecture: Conjecture 3.3.3 Let ft be any simply-connected domain, not necessarily convex. Then the gradient WRmQ of the regular part of the modified Green's function given in (3.84b) has a unique 127 root inside fi. Thus, there is a unique equilibrium location of a one-spike solution of (3.16). In experiments 3 and 4 of §3.4 we consider another example of a non-convex domain that adds further support to this conjecture. To illustrate the novelty of our conjecture, we consider a similar problem for the conventional Green's function Gd with Dirichlet boundary conditions satisfying AGd = -S(x - x 0 ) x e , (3.125a) G<*=0, xedQ,. (3.125b) The regular part of Gd and its gradient are defined by Rd(x, x 0 ) = Gd(x, x 0 ) + ^ - In |x - x 0 | , ' VRd0 = V i ? d ( x , x 0 ) | x = x 0 • (3.126) It was shown in [30] that - -h (rq^F + (M) • ( 3 1 2 7 ) where xo = f{zo) (compare this result with Theorem 3.3.1). Unlike computing the modified Green's function RMQ with Neumann boundary conditions, no knowledge of the singularities of f(z) outside the unit disk is required to compute RDQ = Rd(xo, XQ). It was shown by several authors (cf. [30], [7]) that for a convex domain f i , the function Rdo is convex. Thus, for convex domains, its gradient has a unique root. However the derivation of this result explicitly uses the convexity of the domain. For non-convex domains generated by the mapping (3.120), it was shown in [30] that \7Rdo can have multiple roots. Thus, the Neumann boundary conditions are essential for Conjecture 3.3.3. From Conjecture 3.3.3 together with (3.85), it follows that for D large enough, there is exactly one possible location for a one-spike equilibrium solution. On the other hand, for a dumbell- shaped domain such as in Fig. 3.4, we know from Proposition 3.2.2, that when D is small enough, the only possible minima of the regular part Ro of the reduced wave Green's function are near the centers of the lobes of the two dumbells. In Appendix 3.5, we show that Ro —+ +oo 128 as xo approaches the boundary of the domain. Hence, Ro must indeed have a minimum inside the domain. B y symmetry, it follows that for D small enough, the centers of both lobes of the dumbell correspond to stable equilibrium locations for the spike dynamics. In addition, it also follows by symmetry and by Proposition 3.2.2 that the origin is an unstable equilibria. Hence, this suggests that a pitchfork bifurcation occurs as D is increased past some critical value D C . As D approaches DC from below, the two equilibria in the lobes of the dumbell should simultaneously merge into the origin. For the non-convex domain of experiment 4 in §3.4, this qualitative description is verified quantitatively by using a boundary element method to compute Ro- 3.4 N u m e r i c a l Exper iments and Discuss ion In this section we perform numerical experiments to verify the results of this chapter. In experiments 1 and 2 we compare the asymptotic formulas (3.17), (3.84) and (3.85) with corresponding full numerical solutions of (3.16). In experiment 3 we provide some further numerical evidence for Conjecture 3.3.3. Finally in experiment 4 we will study the effect of the shape of the domain and the constant D on the the existence of spike equilibria location. The asymptotic results require us to compute Ro, Rmo, and their gradients. For experiments 1 and 2 we restrict ourselves to a square domain. For this case Ro can be computed using the method of images solution (3.58). Equation (3.58) also works well for D = O ( l ) . However, note that the number of terms needed in (3.58) to achieve a specified error bound is directly proportional to D. Therefore the time cost is given by 0(D) (the storage cost being constant). To compute RQ on a non-square domain (or on a square domain when D is large) as well as to compute Rmo on any domain to which Theorem 3.3.1 does not apply, we have adopted a Boundary Element algorithm as described in §8.5 of [4]. Details are given in §3.4.1 To compare with our asymptotic results we use a finite element method to solve (3.16). A standard numerical finite element method code does not perform well over long time intervals due to the very slow movement of the spike and the very steep gradients near the core of 129 the spike. To overcome this, we have collaborated with Neil Carlson 'who kindly provided us with his moving-mesh program, mfe2ds with adaptive time step (cf. [8], [9]). This has reduced the computer time dramatically, because much fewer mesh nodes or time steps were required. However, the solution obtained from the current version of mf e2ds, tends to deviate from the expected solution after a long time period. This is because as the spike moves, it moves the mesh along with it, until eventually the mesh is overstretched (see Fig. 3.5). In spite of this limitation for long-time computations, full numerical results for (3.16) are computed using mf e2ds. 3.4.1 Boundary E lement method We now describe the Boundary Element method (BEM) used to compute the regular part R(x, xo). We write: G(x,x0) = V(x,xo) + R(x,x0), V(X,XQ) =-^-K0(X\x - XQ\).. (3.128) ZlT where Ko(z) is the modified Bessel function of order zero. From (3.10) we obtain R(x, xo) = R(x, x0) + ^- [log \x - x 0 | + Ko (A|x - xQ\)] (3.129) Using the local behavior of KQ(Z), (3.56b) we obtain R(x,x0) = R(x, x0) - (log 2 - 7 - log A) + o( l ) , as x -> x0 , (3.130) where 7 is Euler's constant. Therefore, since V R(x;xo)\x=xo = VR(X;XQ)\X=X0, it suffices to compute R(X,XQ). Substituting (3.128) into (3.10), we obtain that R(x,£) satisfies AR(x,£) - A 2 i ? ( x , 0 = 0, xett, (3.131a) dnR(x,Z) = -dnV(x,£), xedfl (3.131b) where A = -j=. The integral representation for R is R(x, 0 = - / G(x, v)dnV(r], £) dS(r,). (3.132a) Jon 130 Using (3.128), this can be written as R(x, 0 = - / R(x, v)dnV(ri, 0 dS(ri) - [ V(x, v)dnV(V, £) dS(rj). (3.132b) Jon Jan Next, we discretize the boundary dQ. into n pieces dQi, • • • ,dQn, and approximate R(x,£) = R(x,£i) for £j € dQ,i, where & is the midpoint of the arc df^. Letting Rj = R(x,£j), we can then approximate (3.132b) by the dense linear system n Rj = E (v-ijRi + bij^j , (3.133a) t=i where Oif=- [ dnV(V,^)dS(r,), bij = - [ V(x,71)dnV(rj,CJ)dS(rl). (3.133b) After calculating the solution to (3.133a), we can determine R(X,XQ) by discretizing (3.132b). This leads to R(x, x0) = - E ( y0> + ^ i ) ^n^(r?i, x 0 ) / i . (3.134) i=l Here ^ is the length of <90i. It remains to compute the coefficients a -̂ and bij in (3.133b). When i ^ j we have ay = - / i 5 n y ( r y i , £ j ) , bij = -liV(x,r]i)dnV(rii,€j). (3.135) The case i = j requires a special treatment because of the logarithmic singularity of the free- space Green's function V. Let r be the radius of curvature of <9fii at and set Kj = 1/r. Let / = l{ be the length of dVt{. Since is small, we may assume that dfij is parametrized for t < 1 as rrffJ) = r(cosi ,sint) , — < £ < , (3.136a) 2r 2r with £ = & = (r,0). (3.136b) The asymptotic behavior V(r), £) ~ — ̂  log |ry — £| + 0(1) as |rj — £| —> 0 yields ~ - S h F & - * - ( 3 1 3 7 ) 131 t = 0 t = 1 0 . 8 1 4 3 2 t = 1 0 0 0 0 . 0 0 O 0.2 0.4 0.6 O.B 1 O 0.2 0.4 0.6 0.8 1 O 0.2 0.4 0.6 O.B 1 Figure 3.5: Numerical solution using the moving mesh method. At the beginning, the mesh vertices concentrate at the spike. Then they move along with the spike, eventually overstretch- ing the mesh geometry and resulting in a loss of precision over a long time period. In this example, e = 0.01, D = 5, and the initial conditions at t = 0 were a = sech(|x — (0.3,0.5)|) and h = 1. Since n = (cost,sint) we calculate (77 — £) - n = r ( l — cost), and \q — £ | 2 = r 2 (2 — 2cost). Hence, from (3.137), 0 n f ( 7 7 , O ~ - ; r - . a s r ^ 0 ' ( 3 - 1 3 8 ) 4 7 r r Therefore, the coefficients an and bn in (3.133b) are a " = £ ^ * > ^ « = ttii^^)^) • (3.139) 3.4.2 Experiment 1: Effect O f e W i t h D = 1. Fig. 3.6 shows the the peak location (x(i),0.5) versus time for a unit box [0, l ] 2 , with D = 1 at several values of e. It shows that the asymptotic approximation (3.17) is very close to the full numerical results when £ = 0.01, and it still gives a reasonable approximation even when e = 0.1. Presumeably, we would have a much closer agreement at e = 0.1 if we had retained higher order terms in the infinite logarithmic expansion of ao, ho, and the adjoint eigenfunction tp, in the derivation of (3.17) from (3.39). From Fig. 3.6 we note that the full numerical solution for x{t) seems to settle at something less than 0.5. This is a numerical artifact of the current implementation of the moving mesh code. We think that this is caused by the over-stretching of the mesh topology. 132 6 10OO 2000' 3000 4000 5000 O 10O 200 300 400 500 Figure 3.6: Movement of the center (x(£),0.5) of a single spike of (3.16) within a unit box [0, l ] 2 versus time t. Here D = 1 and e = 0.01,0.05,0.01 as shown. The solid lines show the asymptotic approximation (3.17). The broken lines show the full numerical solution computed using mfe2ds. The figure on the right compares the asymptotic and numerical results on a smaller time interval than the figure on the left. 3.4.3 Experiment 2: Effect Of D With e = 0.01 Fig. 3.7 shows the peak location (x(£),0.5) versus time for a unit box [0, l ] 2 , with e = 0.01 and D = 1,3,5. For each value of D, the full numerical solution as well as the asymptotic approximations (3.17), (3.84) and (3.85) are shown. While we assumed in the derivation of (3.84) that D 3> 1, the simulation shows that even for D = 1, the approximation (3.84) is rather good. Notice that the approximation (3.85), which does not involve any terms involving — Ine and the modified Green's function in the denominator, provides a significantly worse approximation to the spike dynamics than either (3.84) or (3.17). This point was mentioned at the end of §2. As before, the numerical solution given by mf e2ds seems to deviate from the expected solution after a long time due to excessive mesh stretching. This effect is especially pronounced for larger D. We are currently working with Neil Carlson to address this problem by combining moving mesh with mesh refinement algorithms. 133 O 20O0 4000 6000 6000 10000 0 1000 2000 3000 4000 5OO0 6000 7000 0 4 0 1OO0 2000 3000 4OO0 5000 D=1 D - 3 D=5 Figure 3.7: Movement of the center (x(i),0.5) of a single spike of (3.16) within a unit box [0, l ] 2 versus time t, with e = 0.01 and D = 1,3, 5. The solid curves show the asymptotic approximation (3.17). The bottom and and top curves show the approximations (3.84) and (3.85), respectively. The diamonds show the results from the full numerical simulation. 3.4.4 E x p e r i m e n t 3 : U n i q u e n e s s O f E q u i l i b r i a F o r L a r g e D We have used a boundary element method to numerically compute VRmo for the non-convex domain Cl shown in Fig. 3.8. There is only one equilibrium solution in Cl and it lies along the imaginary axis as indicated in the figure caption. This provides more evidence for Conjecture 3.3.3. 3.4.5 E x p e r i m e n t 4 : A P i t c h f o r k B i f u r c a t i o n . In this experiment we consider the dumbell-shaped domain Cl = f(B) as given by (3.120), and we study the effect of the neck width as well as D on the roots of VRQ. To compute Vi?o we used the boundary element method described in §3.4.1, discretising the boundary into 200 elements. Since Cl is symmetric we look for spike equilibria that are along the x-axis. When the dumbbell shape-parameter is b = 1.2, and for the values of A as shown, in Fig. 3.9a we plot RX along the segment of the positive x-axis that lies within the dumbbell. Notice that there are either one, two, or three spike equilibria on x > 0 depending on the range of A = D - 1 / 2 . The resulting subcritical pitchfork bifurcation diagram for the spike equilibria is shown in Fig. 3.9b. Our computations show that there is a pitchfork bifurcation at A « .3.74, where A = D - 1 / 2 . Furthermore, there is a fold-point bifurcation of spike equilibria when A « 2.59. The spike at 134 Figure 3.8: Plot of VRmo for a non-convex domain whose boundary is given by (x, y) = (s in 2 2i + \ sin£)(cos(£), s in(Tj)), t e [0, rr]. Its center of mass is at about (0,0.464), which lies outside the domain. The resulting vector field has only one equilibrium, at approximately (0, 0.2). The discretization of the boundary that was used for the boundary element method is also shown. the origin is stable when A < 3.74, and is unstable for A > 3.74. In Fig. 3.9b, the upper branch of spike equilibria is stable, while the middle branch is unstable. A subcritical bifurcation diagram of this type has not computed previously. Notice that as A —> oo (D —> 0), the upper branch corresponds to stable spike equilibria that tend to the lobes of the dumbbell as D —> 0. 0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 x lambda (a) Rx versus x for x > 0 (b) x versus A = D l/2 Figure 3.9: (a) Plot of Rx when x is along the positive real axis, for the values of A as indicated. The dumbbell shape-parameter is b = 1.2. (b) The subcritical bifurcation diagram of the roots of R^ = 0 versus A = D~1/2 when b = 1.2. Next, we investigate numerically the effect of changing the dumbbell shape-parameter 6'. In Fig. 3.10 we plot the numerically computed bifurcation diagram of spike equilibria for nine 135 different values of b. The leftmost curve in this figure correspond to b = 1.15. Successive curves, from left to right in Fig. 3.10, correspond to an increment in b of 0.05. Qualitatively, we observe from this figure that the equilibrium spike at the origin has a subcritical bifurcation in A only when 1 < b < bc. For b > bC) the origin has a more conventional supercritical pitchfork bifurcation. We estimate numerically that bc « 1.4. Since for b —> 1 + the domain Q reduces to the union of two disconnected circles each of radius 1/2, Fig. 3.10 suggests that the bifurcation of spike equilibria at the origin is subcritical when the neck of the dumbbell is sufficiently narrow, and is supercritical when the domain is close to a unit circle (b large). It would be interesting to investigate more generally whether certain broad classes of dumbbell-shaped domains with thin necks will always yield subcritical pitchfork bifurcations for a one-spike equilibrium of (3.1) when D = 0(1). We remark that the stability properties of the branches of equilibria in Fig. 3.10 are precisely the same as described previously for Fig. 3.10. For each b > 1, there is still a stable spike equilibrium that tends to a lobe of the dumbbell as A —> oo (D —» 0). 0.4 0.2 x o- -0.2- -0.4- , 4 ' ' 5 ' ~ 6 lambda Figure 3.10: Plot of the bifurcation diagram for the spike equilibria versus A = D~1'2 for various values of the dumbbell shape-parameter b. The curves from left to right correspond to b = 1.15, b = 1.2, 6 = 1.25, b = 1.3, b = 1.35, b = 1.4, b = 1.45, 6 = 1.5, and b = 1.55. In Fig. 3.11a we plot the bifurcation diagram in the A versus b parameter plane. From this figure we observe that when b > 1.4, there is only one bifurcation value of A, and it corresponds to the pitchfork bifurcation point for the equilibrium XQ = 0. For 1.15 < b < 1.4, there are two bifurcation values for A. The larger value of A corresponds to the pitchfork bifurcation value, and 136 the smaller value of A corresponds to the fold-point value where the middle and upper branches of spike equilibria associated with the subcritical bifurcation coincide. Finally, in Fig. 3.11b, we plot the fold-point value for the spike equilibria as a function of b for 1.15 < b < 1.4. The non-smoothness of this curve reflects the fact that, due to computational resource limitations, we only had nine data points to fit with a spline interpolation. lambda 1.2 1.3 1.4 1.5 1.6 1.7 1.8 ~ ° - 3 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 b b (a) A = D~1/2 versus b (b) Fold-point locations Xf versus b Figure 3.11: (a) Curves of A versus b where the spike equilibria have either a pitchfork bifurcation or a fold-point bifurcation, (b) Locations xf of the fold-point bifurcation versus b. 3.4.6 Discussion There are several open problems that await a rigorous proof. A main conjecture 3.3.3, is that the gradient V i ? m of the regular part of the modified Green's function with Neumann boundary conditions has a unique root in an arbitrary, possibly non-convex, simply-connected bounded domain. In contrast, as was shown in [30], this is not true if Dirichlet boundary conditions are used instead. Many properties of the gradient of the regular part of the Green's function for the Laplacian with Dirichlet boundary condition have been given in the survey [4]. The uniqueness of a root to this gradient with Dirichlet boundary conditions in a convex domain is established in [30] and [7]. Our conjecture shows that further work is needed to understand the properties of the regular part of the Green's function associated with a Neumann boundary condition. A second conjecture, based on §6, is that the zeroes of the gradient VR of the reduced wave Green's function will have a subcritical bifurcation with respect to A = D - 1 / 2 in a dumbbell- shaped domain, whenever the neck of the dumbbell is sufficiently thin. Alternatively, we con- jecture that the zeroes of V i ? will have a supercritical bifurcation in A when a dumbbell-shaped 137 domain is sufficiently close to a circular domain. 3.5 A p p e n d i x : T h e Behav ior O f R0 O n T h e B o u n d a r y T h e o r e m 3.5.1 Suppose dQ is C2 smooth. Let x E dQ and let xo(d) = x — dh be the point a distance d away from x and dQ. Then there exist positive constants C\,C2,£ such that R(x0, x0) > C i ln - + C*2 , (3.140) for all d < £ sufficiently small, where R is given by (3.18). Proof . When Q is convex, this theorem was proven in the Appendix of [84]. However, the convexity assumption was critical in their proof. Our proof below does not require this assump- tion. The proof in [84] utilized a boundary integral representation of R. We use the comparison principle instead. It suffices to prove this result for R replaced by R. From (3.10) and (3.54), R satisfies AR(x, xo) — X2R(x, xo) = 0, x E Q ; dnR(x, xo) = — dnV(\x — xo\), x e dQ, where A = 1/VD~, and V(r) = ±-K0(\r). By rotating and translating, we may assume that XQ = (d, 0) and x = 0. We parametrize dQ by its arclength x(s) with x(0) = x . We let xr0 = ( -d , 0) be the reflection of xo in x and define K be the curvature of dQ at x . Step 1. Show that d n v ( \ x - 4 | ) - c < -dnv(\x-xo\), (3.141) for some constant C , for all s, d < e, and e small enough. < 138 The case K < 0 is more involved. We have, x(s) = {^s2 + o{s3),s + o(s3)), h=(-l + o{s2),Ks + o(s2)), where K is the curvature at x . Here and below, o(sp, dq) is some function such that o(s p, dq) C(\s\p + \d\q) for some constant C and for all |s|, \d\ < e. For x = x(s), we have (a: - x0(d), n) = | s 2 - d + o(s2)d + o(s3), and |x - x0\2 = d 2 + s 2 ( l - Kd) + o(s3). Next, we calculate that -8nV(\x - sol) = ^ ~ X ° ' | 2 n ) + - xo\) by (3.3b), Z7T | X — X\L _ 1 f s 2 _ ^ + o ( s 2 ^ + o ( s 3 ) 2TT d 2 + S 2(1 - «;d) + O(S 3 ) + 0 ^ S ' ^ ' ! f s 2 _ d + o ( s 2 ) d + o ( s 3 } / 0 ( S 3 ) V , 1 - 1 2 - T T 2 7 1 ^ + • • • + ° ( S > D ) > 27T d 2 + s2 In a similar way, we get 2TT d 2 + s 2 ( l - K d ) V d 2 + s 2 ( l - K d ) =h { d 2 + s 2 \ i - K d ) + o ( s ) ) ( 1 + o ( s ) ) + o ( 5 ' d ) ' _ 1 %s2~d d ~ 2lT d2 + S\l - Kd)+ ° [ S ) d? + S2{1 - Kd)+ °{S> ^ = ^ f e ^ ( 1 + + ° ( S ) aWTV2 + °(S' d ) ' 1 f s 2 - d o(sd) . JN = 2 ^ d ^ + d ^ + ° ( s ' d ) ' - —^ - + o(l) + o(s,d). dnV{\x - f0\) = i - j J ' a 2 d + o(l) + o(S, d). Therefore, 1 § s 2 - d 2TT d 2 + s2 -dnV(\x - sol) > 7 ^ V - T + C i > 1 - f s 2 W I * - - 5 l ) < ^ - ^ - + c a 139 for some constants C\, C2 independent of d,s. Notice that Hence, - § s 2 - d p 2 ~ d d 2 + S 2 d 2 + s 2 + \ K \ dnV{\x - x0\) + + C l - C 2 < -dnV(\x - sol), which completes the proof of Step 1. Step 2. By the compactness of the set (dQ\{x(s), \s\ < e}) x {xo(d), 0 < d < e}, the continuity of V(\x — xo\) on this set, and (3.141), it follows that (3.141) holds for all x 6 dfl and all d < e if the constant C is large enough. Step 3. Let u(x) be the solution of A i i — X2u = 0, i 6 i l ; dnu = — C, x € dQ,, where C is as in step 2. Then v = ^ ( | x — XQ|) + u also satisfies Av — \2v = 0 and dnv(x) < dnR(x, xo) for x e cTi and any d, e. B y the maximum principle, it follows that v(x) < R(x, xo) for x € Q- Recalling (3.56b) completes the proof. • 140 Chapter 4 Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model and near-boundary spikes In this chapter we continue the study of (3.1), but in the regime where the diffusivity coefficient D becomes exponentially large. The motivation for this is as follows. It is known (see for instance [37]) that when D = oo, a single interior spike will drift exponentially slowly towards the nearest boundary. As such, there is no stable spike equilibria in the interior of the domain. On the other hand, as shown in §3, when D is large enough (but not not exponentially large), there is a unique stable interior equilibrium. The reason for this apparent discrepancy is that the motion of the spike is driven by the superposition of two fields: an exponentially small boundary field which is of 0(e2e~2dl£) where d is the distance of the spike to the nearest boundary, and the field due to the Green's function whose strength is of O ( ^ ) . Thus the boundary only starts to have an effect when D = 0(e~2e2dl£). In such a regime, there is a transition between the dynamics driven by the Green's function as discussed in §3, and the dynamics driven by the boundary, as discussed in [37]. The primary purpose of this chapter is to study this transition and the bifurcations that occur therein. As an example of the complicated bifurcation structure that results, we again consider the dumbell-shaped domain discussed in §3. In Figure 4.1 we schematically illustrate the various bifurcations that occur as D is decreased from infinity to 0(e2). Note that the unstable equi- libria at D = oo undergoes a complicated bifurcation structure, resulting in a single stable equilibria at the origin as D is decreased to 1 <C D <C exp(d/e). 141 (a) D = oo (b) D = 0(e2e2d'E) Figure 4.1: Schematic representations of vector fields and the bifurcations that occur for interior spike solutions in a dumbbell-shaped domain as D is decreased from oo to 0(e2). Stable equilibria, saddle points, and unstable equilibria correspond to black dots, hatched dots, and white dots, respectively, (a) For the shadow system all equilibria are unstable and correspond to critical points of the distance function from the center of the spike to the boundary, (b) In the intermediate regime, a competition between VRm and the distance function determines the locations of an equilibrium interior spike, (c) For D large, but not exponentially large, there is a unique (stable) spike equilibrium inside the domain located where VRm = 0. (d) For D = 0(1) and 0(e2) « ! ) « O( l ) , an interior spike is located where V i ? = 0. (e) A n intricate bifurcation structure may be present in the intermediate regime between (c) and (f). (f) On the range 0(e 2 ) « D « 0(1), R can be approximated by the distance function. Thus, in (f) the field is the the same as (a), except that the direction is reversed. This transition, from a stable interior spike when D is large, to the dynamics driven by the nearest boundary in the case D = oo, is made possible by the existence of what we call near boundary spikes, in the transition regime when D is exponentially large. More precisely, we show that there exists a spike equilibria location, at a distance o from the boundary, whenever £ <S cr < 1 and where C (p) is some positive constant depending only on p. We find that such a near-boundary equilibria is always unstable in the radial direction. Its stability in the tangential direction, as well as its location, depends on the behaviour of the modified Green's function at the boundary. For the dumbell example above, we show that there are four such equilibria, two of them stable and two unstable, as shown on Figure 4.2. We further conjecture that these near-boundary spikes are related to the boundary spike solu- tions. For the case D = oo, such solution was derived in [38], There, it was found that a single boundary spike is located at the minimum of the curvature of the boundary. For the case of exponentially large D, we expect that the Green's function on the boundary will also play a role in determining the boundary spike location. Figure 4.2: Plot of local vector field associated with near-boundary spikes for the transition regime (4.1). The results of this chapter first appeared in [47]. (4.1) 143 4 . 1 T h e dynamics of a spike for exponent ia l ly large D We now outline the derivation of the equation of motion for a spike solution to (3.1) in ft 6 M 2 . We will use the following scaling of (3.1): at = ezAa - a + — , X <E ft, t>0, (4.2a) tht = DAh - h + e~~2—-, x e f t , t>0, (4.2b) ns dna = dnh = 0, x e dfi. (4.2c) For the shadow problem where D = co, the spike motion is metastable and is determined by the exponentially weak interaction between the far-field behavior of the spike and the boundary 80, (cf. [37]). For D S> 1, but with D not exponentially large as e —» 0, the exponentially weak interaction of the spike with the boundary is insignificant in comparison to the local behavior of the inhibitor field near x = XQ, which is determined by the Green's function Gm of (3.81). When D is exponentially large, the dynamics of XQ is determined by a competition between the exponentially weak interaction of the spike with the boundary and the gradient of the regular part of the modified Green's function. Since this competition is essentially a superposition of the previous results in [37] and §3 (also [46]), we only outline the key steps in the derivation of the dynamics for XQ. Let xo € ft, with dist(xo; dft) 2> 0(e). For D ^> 1, we get from (4.2) that h ~ H on ft, and that a ~ Ww (e^\x - x6\) , 7 = —~— • (4.3) i p - 1 Here the radially symmetric solution w(p), with p = \y\, satisfies w" + -w - w + wp = 0, p > 0 , ' (4.4a) P u i ( 0 ) > 0 , u/(0) = 0; w(p) - ap~1/2e~p, as p -> oo, (4.4b) for some a > 0. Substituting (4.3) and the expansion h = Ti + h\/D + • • • into (4.2b), we find 144 that hi satisfies Ah1 = H- bmWm-s5(x -x0), i 6 Q ; dnh = 0, x € dtt, (4.5) where bm = JR2 wm dy. This problem has a solution only when Ti satisfies f-(im-(s+1) = \Q\/bm, where | f i | is the area of £1. The solution for hi is hi = H\fl\Gm, where Gm satisfies (3.81). To determine an equation of motion for xo, we substitute a = 'H'yw{y)+v and h = 7i+hi/D+- • • into (4.2a), where y = e _ 1 [x — xo(t)]. Assuming that v <C 1, we then obtain the following quasi steady-state problem for v: L£v = e2Av-v + pyjP^v =  g|°lH 7 wPGm - g - 1 ^ ~ X°] • x'0 Ww , x e Q , (4.6a) D \x — xo| dnv = -Wdnw, x e f f i . (4.6b) As shown in [37], the eigenvalue problem Le(f> = Xo4> in fi, together with dncp = 0 on 50 , has two exponentially small eigenvalues A;, with i = 1,2. The corresponding eigenfunctions have the boundary layer form <f>i ~ dx.w + fa . (4.7) Here x = (xi ,X2), and <J3J is a boundary layer function localized near <9Q that allows the no-flux condition for fa on 80. to be satisfied. Multiplying (4.6a) by fa, and integrating over Q, we let e —* 0 to get the limiting solvability condition e-lx0- [ ^~X°}w'dXiwdx=^- [ GmwpdXiwdx+ [ e2fadnwdS, (4.8) for i = 1,2. Since w is localized near x = xo, we let y = £ _ 1 ( x — xo) in (4.8), and use the following local behavior for Gm: Gm(x0 +ey,xQ) =-^\og(e\y\) + Rm(xo,x0) + ey-VRrnO + 0(\y\2). (4.9) 27T Here VRmQ = VRm\x=X0. In this way, we obtain from (4.8) that x'o-[ r l h ' ( M ) l 2 ^ = T^TITvî • / ^ K + 1 ( y ) ] ' ^ + / e ^ w d s , (4.10) JR 2 M L J -D(P+1) JR 2 M Jan for i = 1,2, where y = (2/1,2/2)- The integrals over IR2 in (4.10) were calculated in (3.48). The integral in (4.10) over dd was calculated in [37] (see §3.3 of [37]). In this way, we obtain the following main result: 145 P r o p o s i t i o n 4.1.1 Let e < 1, and assume that x0 G Cl with dist(x0; 80) » 0(e). Then, for D exponentially large as e —> 0, £/ie spz&e location XQ satisfies the differential equation dxo 2eq dt p - 1 c? ( p - \ \ j e\0\ J TT^RmO (4.11a) L27T/H q J" D where the vector boundary integral J is defined by J= f - e ~ 2 r / £ (1 + f -h)f -ndS . (4.11b) Here r = \x — xo\, and r = (x — xo)/r. The constant a is defined in (4-4°) and Q is defined by. f°° r 12 0= p w'(p) dp. ' (4.12) Jo L J The dynamics (4.11) for XQ expresses a competition between V i ? m o , inherited from the local inhibitor field, and the boundary integral J , representing the exponentially weak interaction between the tail of the spike and the boundary 80. The dynamics depends on the constants a and 6, defined in (4.4b) and (4.12). For several values of p, these constants were computed numerically in [73], with the result a = 10.80, 8 = 2.47, p = 2; a = 3.50, 3̂ = 1.86, p = 3; (4.13) a = 2.12, /3 = 1.50, p = 4. When D » 1, but with D not exponentially large as e —> 0, (4.11) reduces to the gradient flow dxo 2e 2o|Q| -d7~ ~W^i) VRm°- ( 4 - 1 4 ) This limiting result was derived independently in [76] and [14], and was also obtained in §3.2.2. The conjecture 3.3.3 is that VRmo = 0 has exactly one root in the interior of an arbitrary, possibly non-convex, bounded and simply-connected domain Cl. 4.2 A R a d i a l l y Symmet r i c D o m a i n : D E x p o n e n t i a l l y Large In this section we analyze the dynamics and equilibria of a spike solution to the Gierer- Meinhardt model (4.2) in a two-dimensional unit disk Si = {x : \x\ < 1} when D is exponentially large. 146 We look for a solution to (4.2) that has an interior spike centered at XQ e ft. Since |ft| = TT, we obtain from (4.11) that the dynamics of the spike satisfies dxo 2eq dt p-l a2 f p — \ \ sir (4.15) >/? V q J D where J was defined in (4.11b), and the constants a and 3 were defined in (4.4b) and (4.12), respectively. For such a ball domain, the gradient of Rmo was calculated previously in [76] as B y symmetry, we need only look for an equilibrium solution to (4.15) on the segment XQ G [0,1) of the positive real axis. To do so, we need Laplace's formula (cf. [86]) valid for e < C 1, / r-lF(r)e-2^ dS ~ V — F(rm) (1 - Kmrm)-1'2 e ^ l * . (4.17a) Jan ^ W / Comparing (4.11b) with (4.17a), we take F{r) = f (1 + f • h)f • n. (4.17b) In (4.17a), rm = dist(xo;9ft), Km is the curvature of dfi at xm, and the sum is taken over all xm € dfi that are closest to XQ. The sign convention is such that KM > 0 if ft is convex at xm. We first suppose that xo 3> 0(e) and that 1 - xo > 0(e). In this case, the point (1,0) on <9ft is the unique point closest to XQ. Using (4.17) with r = (1,0), n = (1,0), and KM = 1, we get for e < 1 that \x0( where i = (1,0). J~2Ucn̂ iJ '"r™t- (418) Next, suppose that XQ > 0 with XQ = O(e). In this case, Laplace's formula (4.17) fails since the asymptotic contribution to J arises from the entire integral over the boundary rather than from a discrete set of points. Parameterizing dfi by x = cost and y = sin£, we calculate for xo <§; 1 that r = 1 — xo cos t + O ( X Q ) ; f = (cos t, sin t) + XQ (— sin 2 t, sin t cos t) + O ( X Q ) ; r • n = 1 + O ( X Q ) . 147 Substituting (4.19) into (4.11b), we obtain to leading order that /•27T J ~ 2 i e- 2 / £ / (cos t) e2e x o c o s * dt. (4.20) Jo Since the modified Bessel function I\ of the first kind of order one has the integral representation (cf. [1]), 1 r2lx J i ( x ) = — / (cos 9) ex c o s 8 d6, (4.21) 2TT JO we obtain that J - 4 7 r i e - 2 / e / 1 (2x 0/e) • (4.22) This expression is valid for xo = 0(e). When xo -C 0(s), the asymptotic evaluation of J is obtained by using the local behavior I\(z) ~ z/2 as z —> 0 in (4.22). This yields, J ~ 4 7 r £ - 1 2 x 0 e " 2 / e . (4.23) Using h(z) ~ (2ixz)-ll2ez in (4.22), we obtain that the far-field form of (4.22) for x 0 > 0(e) agrees with the leading order behavior of (4.18) as xo —> 0. Combining (4.18) and (4.22), we can write a uniformly valid leading order approximation to J as 4 7 r 7 ^ ~ 2 / e J ~ n h (2x 0/e) • (4.24) V l - xo This formula is valid for xo > 0, with 1 — xo 2> 0(e). The origin x 0 = 0 is an equilibrium point for (4.15) for any D > 0. Using (4.23) and (4.16) we can write the local behavior for (4.15) when XQ <C 1 as dx0 2e2q ( D \ — - 1 ) x 0 , (4.25a) dt ( p - l ) D ^ j where Dc is defined by Hence, the equilibrium xo = 0 is unstable when D > Dc, and is stable when D < Dc. The constants a and 8 in (4.25b) are given for various p in (4.13). Substituting (4.16) and (4.24) into (4.15), we obtain the following main result: 148 Proposition 3.1: Let £ < 1 and assume that the spike location XQ within the unit ball is along the segment of the real axis satisfying XQ > 0 with 1 — XQ 3> 0(e). Then, the trajectory Xo(t) of an interior spike solution satisfies dxQ ~~dt 2e2q (p-l)D D Dc H(x0) eIi(2xQ/e) V 1 - xo (4.26a) where Dc is defined in (4- 25b), and H(XQ) is defined by H(xQ) = - 1 ~ + X°^11 ^2x°/£) (4.26b) x 0 (1 - x2/2) Since H(0) = 1, the equilibrium location XQ = 0 is stable when D < Dc and is unstable when D > Dc. On the range xo > 0 with 1 — xo S> 0(e), for each D < Dc there is a unique unstable equilibrium solution to (4-26a) satisfying Dr D = H(x0). (4.27) The local behavior of the bifurcating branch, obtained by setting y = xo/e in (4-27), is given by D h(2y) (4.28) Figure 4.3: Plot of the local bifurcation diagram (4.28) where y = XQ/E. In Fig. 4.3 we plot the local bifurcation behavior (4.28). Qualitatively, we see that, except within an 0(e) neighborhood of xo = 1 where (4.27) is not valid, (4.27) shows that the ratio Dc/D increases as xo increases. Therefore, the unstable spike, which bifurcated from xo = 0 at 149 the value D = Dc, moves towards the boundary of the circle as D decreases below Dc. From (4.25b), the critical value Dc is exponentially large as e —* 0 and depends on the parameters a and Q given numerically in (4.13). Since our analysis has only considered interior spike solutions that interact exponentially weakly with the boundary, we cannot describe the process by which the unstable interior spike merges onto the boundary of the unit ball at some further critical value of D. However, using (4.27) we can give an estimate of the value of D for which a spike approaches to within an 0(a) neighborhood of the boundary, where e C a « 1. Let XQ = 1 — a in (4.27). A simple calculation using (4.25b), (4.26b), and the large argument expansion of h(z), yields Since we have specified XQ G (0,1] at the outset of the analysis, our bifurcation analysis cannot determinine the direction in which the equilibrium spike moves towards the boundary as D is. decreased. This degeneracy in the fundamental problem can be broken by a slight perturbation in the shape of the domain. A resulting imperfection senstitivity analysis would presumeably be able to resolve the degeneracy and determine a unique direction for the bifurcating spike. 4.3 A Dumbbe l l -Shaped D o m a i n : D Exponen t i a l l y Large When D is exponentially large, we now analyze the dynamics and equilibria of a one-spike solution to (4.2) with r = 0 in a one-parameter family of dumbbell-shaped domains introduced in [30] and studied further in §3.3.1. There, we considered a family of domains given by Cl = f(B) where B = {z : \z\ < 1} is the unit disk in a complex plane and Here b is real and b > 1. The resulting domain Cl = f(B) is shown in Figure 3.4 for several values of b. Notice that Cl —> B as 6 —> oo. Moreover, as S = b — 1—> 0+, Si approaches the union of two circles centered at (±1 /2 ,0 ) , with radius 1/2, that are connected by a thin neck region of width 25 + 0(52). This class of dumbbell-shaped domains is symmetric with respect (4.29) w = f(z) = (1 - b2)z z 2 - b 2 (4.30) 150 to both the x and y axes. Therefore, when D = oo, and for a certain range of 6, we expect that there will be three equilibrium one-spike solutions, one centered at the origin and the other two centered on the x-axis in the lobes of the dumbbell. It is easy to show that Q is non-convex only when 1 < b < bc = 1' + y/2- Let xo be the location of a one-spike solution to (4.2) in f i , with pre-image point ZQ € B satisfying xo = f(zo). Recall from 3.3.1 that V i ? m o as defined in (4.9) is given by: V i ^ o = 5M, (4.31a) /Oo) where If z0 2b2z0 b2z0 VsOo) = — r - T o - z a —U + 2TT \1-\ZO\2 Z%-& z2b2-l (fr4 - l ) 2 ( k o j 2 -l)(z0 + b2zQ)(z2 + b2) (64 + l ) ( z ^ _ 1 ) ( z 2 _ 6 2 ) ( ^ _ 6 2 ) 2 (4.31b) and / ( z 0 ) = ( b 2 - l ) ( f + 6 2 2 ) 2 , x 0 = / 0 o ) . (4.31c) \zo ~ 0 ) In (4.31), we interpret vectors as complex numbers so that V i ? m o = dxRmQ + idyRmQ. The area \Q,\ of fl, which is needed below, was also derived in 3.3.1 to be Note that \ft\ —-> TT as b —> oo, and —> 7r/2 as b 1 + , when 0, reduces to two circles each of radius 1/2. The dynamics and equilibria of a one-spike solution to (4.2) in ft is obtained by substituting (4.31) and (4.32) into (4.11). We show below that the bifurcation behavior of equilibria to (4.11) is as sketched in Figure 4.1. In §4.3.1 we analyze this behavior for the equilibrium spike located at the origin (0,0) in the neck region of the dumbbell. In §4.3.2 we analyze equilibrium spikes in the lobes of the dumbbell. 4.3.1 The Neck Reg ion of the D u m b b e l l When D = oo, the equilibrium spike solution at the origin is unstable. However, as we show below, as D decreases below some critical value Dc this equilibrium solution regains its stability, 151 a n d n e w u n s t a b l e e q u i l i b r i a a p p e a r a t t h e p o i n t s (0, ± y o ) i n O for s o m e yo > 0. T h e s e e q u i l i b r i a m o v e a l o n g t h e v e r t i c a l a x i s t o w a r d s t h e b o u n d a r y 80 as D is dec reased b e l o w Dc (see F i g u r e 4 .1 (b ) ) . T o a n a l y z e t h e s p i k e b e h a v i o r , we m u s t c a l c u l a t e t h e i n t e g r a l J i n (4.11) a s y m p t o t i c a l l y . T o d o so, we p a r a m e t e r i z e 80 b y l e t t i n g z = elt, for —K < t < ir, a n d w(t) = / (eu) = £ (£ ) + ir)(t). U s i n g (4.30) , we c a l c u l a t e ( 6 2 - l ) 2 c o s £ v(t) = (b4 - l ) s i n t (4.33) b4 + 1 - 2b2cos2£ ' / w 6 4 + 1 - 2b2cos2£ ' F o r a s p i k e l o c a t e d at (0, yo) i n O, w i t h yo > 0 b u t s m a l l , t he d o m i n a n t c o n t r i b u t i o n t o J ar ises f r o m t h e p o i n t s c o r r e s p o n d i n g t o £ ' = ± 7 r / 2 , l a b e l l e d b y (0, ± y m ) , w h e r e y^ 6 2 + l (4.34) A s i m p l e c a l c u l a t i o n u s i n g (4.33) s h o w s t h a t t h e c u r v a t u r e Km o f 80 at (0, ±ym) is K"m. — [r/' 2 + r 2] 3 / 2 b 2 + l t = ± 7 T - / 2 V 6 2 — 1 8 6 2 ( 6 2 + 1 ) ' (4.35) B y s y m m e t r y , t h e v e c t o r i n t e g r a l J i n (4.11) has the f o r m J = (0, J 2 ) . W h e n y 0 < y™ - £, we eva lua t e t h e i n t e g r a l J a s y m p t o t i c a l l y i n (4.11) for e —> 0, t o o b t a i n J 2 ~ 2%Ar¥e- 2 y" l/ £ =22/o/e -2yo/e v / ^m ^ l ~ K m r m i ) ^ / r m 2 ( l - Kmrmq) w h e r e r m i = ym-yo, rm2 = ym + yo •• W h e n |yo | = 0(e), we c a l c u l a t e Jo ~ 4 7T£ ym (1 K"myn 1/2 e - 2 W m / e s i n h (2y 0/e) . (4.36a) (4 .36b) (4.37) N e x t , we c a l c u l a t e V i ? m o nea r t h e o r i g i n . L e t z0 = IVQ € a n d two = i y o € O. S i n c e u ; = f(z), we c a l c u l a t e t h a t (ft2 - 1) 2 y 0 6 2 - l 2 y 0 - 6 2 1/2 (4.38a) 152 and y o ~ / ' ( 0 > 0 , where / (O) = ^ ^ , as y 0 - + 0 . (4.38b). Using (4.32) and (4.38b), we can calculate \Cl\dyRmo in (4.31) in terms of yo for |yo| C 1- A simple, but lengthy, calculation yields that. \n\dyRr,* = yG(b), G(b) = 2[̂ ~f)2 [266 + 364 + 2b2 - l] . (4.39) Substituting (4.37) and (4.39) into (4.11a), we obtain the following main result: P r o p o s i t i o n 4.3.1 Let e « 1 and assume that the spike location (0, yo) on the y-axis satisfies yo = 0(e). Then, the local trajectory yo(t) satisfies ^ ~ e1'2 po e" W « dt sinh (2y 0/g) _ L\ (2y 0/e) D yo, (4.40a) where Dc and po satisfy D c = 3 ( p ^ i ) ( ? ) 1 / 2 [ y m ( i • K m y ^ 1 ' 2 $ ( b ) e 2 y m / £ > ( 4 - 4 ° b ) /"0 = ^ 5 [2/m (1 - Kmym)}~l/2 • (4.40c) When D > D C ; yo = 0 is i/ie unique, and unstable, equilibrium solution for (J^.^Oa). For D < Dc, yo = 0 is stable, and there are two unstable equilibria with \yo\ = 0(e), satisfying 2( D sinh (2C) Dc ' C = y 0 / £ . (4.41) The bifurcation value Dc depends on the dumbbell shape-parameter 6, and on a and 6 defined in (4.4b) and (4.12), respectively. The values a and 6 were computed numerically in (4.13) for a few exponents p. In the limiting case b — 1 = S —> 0 + , we calculate from (4.34), (4.35), and (4.39), that ym-*S, Km^-5-\ 0->3<T78, as <5 ^ 0+. (4.42) Substituting (4.42) into (4.40b), we obtain 153 This limiting formula for the bifurcation point is valid for 0 < e <C 5 <§; 1. Next, we determine the global bifurcation branch for yo > 0 with 0(e) <C yo <C y m — O(e). We set zo = ivo in (4.31b) with 0 < VQ < 1. From (4.31b) and (4.32), we then obtain \Cl\dyRmo = voxi{b;v0), (4.44) where x/(°!' uo) is defined by (64 + l ) ( 0 2 - l ) (b2 + v2)2 Xl(b]v0) =- 2(64 - l ) 2 (b2 - v2) (1 + b2) + 2b2 ( 6 4 - l ) 2 ( 6 2 - l ) ( l - ^ 2 ) ( 6 2 - ^ 2 ) (4.45) _{l-v2)(l + b2v2) v$-b* (64 + l ) ( l + 6 2 V 2 ) (6 2 + i ; 2 ) 3 Here i>0 = vo(yo) i s given by (4.38a). Comparing (4.44) with the local behavior (4.39), and using yo ~ / ' (0) u o for VQ -C 1, it follows that X / ( 6 ; 0 ) = G(b)f'(P), (4.46) where Q(b) is defined in (4.39). As v0 —>• 1", or equivalently y0 —> y~, dyRm0 —> +oo. From (4.45), we calculate as i>o (4.47) On the range y0 > 0 with O(e) <C y 0 < y m - O(e), the term in (4.36a) proportional to e ~ 2 y o / £ can be neglected. To determine the dynamics of y 0 on this range, we substitute (4.36a), and (4.45), into (4.11). After some algebra, we get dyo dt e ll2poe-2y™/e (C(ym)\ e 2yo/f _ Dc (xi(b;vQ) \C(rm)J (Avo/e) D \ Q{b) (4.48a) where rm = ym — y 0 . Here Dc and no are defined in (4.40), and the function C(s) is defined by C(s) = [s(l - Kms)f2 . (4.48b) The equilibria of (4.48a) satisfy Using the formula (4.38a) for v0 = v0(y0), we can write D/Dc as a function of y 0 . For e < 1, the right-hand side of (4.49) is a decreasing function of y 0 . Thus, D is a decreasing function of yo- 154 4.3.2 T h e Lobe Reg ion of the Dumbbe l l Next, we analyze the behavior of an equilibrium spike in the right lobe C of the dumbbell as D is decreased from infinity. Let V, denote the largest inscribed circle within C. Let x m and r m denote the center and radius of 71, respectively. For b = 1.8, in Fig. 4.4 we show the right lobe C and the largest inscribed circle 71, with center on the x-axis. When D = oo, the location XQ of an equilibrium spike satisfies J = 0, where J is defined in (4.11). To leading order as e —-> 0, XQ ~ X j n satisfies J = 0. The correction term is of 0(e) and may also be computed as in [73]. Therefore, to leading order, it suffices to determine x;„. Figure 4.4: The domain Q with b = 1.8 and the largest inscribed circle 71 inside its right lobe C. The largest inscribed circle 7Z makes two-point contact with dCl at xc = (£c, ±nc), with rjc > 0. These points are such that the normal to dd at xc is parallel to the y-axis. From (4.33), this implies that r/' = 0. We write (4.33) as (b2 - I)2 (bA -I) £(t) = ^ '— cos t, T)(t) = '- sin t, (4.50a) X X where X = b4 + 1 - 2b2 cos(2rj) = (b2 + l ) 2 - 46 2 cos 21. (4.50b) Setting 77 = 0, we get sin21 = ^ 2 , or cos t = 0. (4.51) Two roots are t = ± 7 r / 2 . Combining (4.51) and (4.50b), we get x = 2(fr2 — l ) 2 , and that the 155 other root satisfies C 0 S * 462 Substituting (4.52) into (4.50), we obtain that the contact points satisfy ( & ± * ) = . ( l [ « M * + i ) ] * ± ^ ) . Hence, x i n = (£ c ,0) , and r i n = r\c. 1 (4.52) (4.53) Figure 4.5: Plot of dxRm0(x,.Q) versus x for b = 1.05, 1.1, 1.2, 1.5, 2, 2.5, oo. The top (bottom) curve corresponds to b = 1.05 (oo), respectively. Note that dxRmQ(x,0) is positive on the positive x-axis. The formula (4.53) is valid only when 6b2 — (64 + 1) > 0. Therefore, we require that 1 < b < bc = 1 + \/2. In order to show, that r j n is the radius of the largest inscribed circle for this range of 6, we must verify that a circle centered at x m will lie strictly inside the domain, and will only touch the boundary at (£ c , ±r]c). This global verification has been done numerically. As b —> b~, we have x j n —• (0,0) and r j n —> l/y/2. Alternatively as b —+ 1 + , we have x m —> ( | , 0) and r-j n —> 1/4. From (4.35) we conclude that the curvature Km of <9f2 at the point x = 0 tends to zero as 6 —> 6~, Moreover, more algebra shows that the domain is convex when b > bc. The convexity of the domain for b > bc explains the nonexistence of a largest inscribed circle in the right lobe of the dumbbell for this range of b. 156 Next, we determine the equilibrium point xo of (4.11a) in the right lobe of the dumbbell when D is exponentially large. In Fig. 4.5, we use (4.31) to plot dxRmQ along the positive x-axis. Note that <9x-Rm,o > 0 for x > 0. This inequality was shown in §3.3.1. Therefore, as D decreases, the equilibrium location (xo,0) tends to the point (1,0) 6 dfi. When D is sufficiently small, the point (1,0) will be the closest point on the boundary to (xo, 0). In this range of D, we conclude from (4.11) that the location of the spike is determined by a balance between dxRmQ and the dominant contribution to the x-component J\ of the integral J obtained from the closet point (1,0) E dfi. From (4.11), this unstable equilibrium satisfies a2 I q \ , e 2TT/3 \ P - 1 / D Calculating J\ asymptotically as in (4.17), we get ire \ 1 / 2 e - 2 ( i - * o ) / e J i = — I S I I ^ ^ O - . (4.54) J TT? . ( 4 - 5 5 ) 1-XoJ [ l - K ^ l - x o ) ] where K\ is the curvature at the point (1,0) given by «i = 1 + ib2/{b2 + l ) 2 . (4.56) 4.4 Spike E q u i l i b r i a Nea r the B o u n d a r y In this section we will compute the leading-order behavior of V i ? m o near the boundary of the dumbbell-shaped domains of §4.3. We then use this formula to analyze equilibrium spike locations near the boundary. We begin with the following result, which describes the behavior of the Green's function near the boundary. P r o p o s i t i o n 4.4.1 Let fl = f(B), where B is the unit ball and f is given by (4-30). Let z be a point on the boundary dB and let x = f(z) be the corresponding point on the image boundary dfi. Let N be the outward pointing normal at x. Let x0 = x-aN, 0 < a < 1, (4.57) and ZQ satisfy XQ = f(zo). Then + 0(a). (4.58) , x N N V-R m o(x 0 ) = h Airo~ 2n\f'(z)\ z2b2 _ {zA + 56 2 z 2 ) _ 1 z2b2 - 1 z4 - 6 4 4 157 Here and below, ab denotes complex multiplication and (a, 6) = Re(ab) will denote a vector dot product. Proof. We calculate N as ^/(e i 4)| l/ '0)l /'(z) ' i dt From (4.57) and (4.59) we have: f(z0) = x0 = f(z) - ~ / z Thus, for 0 < a <ti 1, we obtain ZQ ~ Z <TZ l/'COl For cr <C 1, we calculate from (4.31b) that = Z 1 Vs(z 0 ) 1 0̂ + z0b2 2TT V l - N l V 2TT V ^ 2 - l ^ O - ^ 4 Substituting (4.61) into (4.62), and using |z| = 1, we obtain 1 / z0 \ _ \f'(z)\z + 0(a) 2TT V l - N : 4?Tcr 1 - cr 2|/'0) + O(or). Next, we note that 1 1 az f'(zo) f'(z) Therefore, using (4.30) for f(z), we get 1 - /'Oo) 27T V l - | Z 0 | V 47TCr/'(z) V 2|/'(z) / : 1 1 + . ... J . _ _ j + 0(a) \f'(z)\f'(z). = N 1 1 zf"(z) \ ^, . + — — ^ - 4 = 4 = I + 0 ( c r ) , 47TCT 8TT|/'(Z)| 4 T T | / ' ( Z ) | / / ( Z ) 1 1 + 47TO- 8TT|/'(Z)| 4TT|/ ' (Z)| - 2 (z2 + 362) z2 + 0 ( a ) . (4.59) (4.60) (4.61) (4.62) (4.63) (4.64) (4.65a) (4.65b) (4.65c) 158 Finally, substituting (4.65c) and the second term of (4.62) into (4.31a), we obtain Vs(z 0 ) V i ? = N ( 1 1/4 2 7 r \ 2 a \f'(z)\ \f'(z) z2b2 (z2 + 3b2) z2 ~4 b4 2z2b2 z2b2 - 1 z4 - b4 + 0(a), N 1 1 + 2TT \2a 4|/'(*)| \f'(z)\ This completes the proof. • z2b2 {z4 + 5b2z2) z2b2 - 1 z4-b4 + 0(a). (4.66a) (4.66b) (4.66c) We now use this proposition to describe the spike dynamics and equilibria near the boundary. Let xo = xo(t) be 0(a) close to the boundary, as defined in (4.57), and assume that £ < C f f C l . We then use Laplace's method to calculate the integral J in (4.11a) as J ~ 2 W — e~2°leN. a (4.67) where J is defined in (4.11b). In deriving (4.67) we have assumed that KG < 1, where K is the curvature of dQ,. Substituting (4.67) and (4.66c) into (4.11a), we obtain dxri 2eq a2 fp—l dt p - 1 [TT6 \ q J V cr e|0| \ N N ™e~2^N + N D | 47TCT 8 7 r | / ' ( z ) | 2TT | / ' ( « ) I \z2b2 - 1 lb2 + 5b2z< z4-b4 (4.68) Note that the first three terms on the right hand side of (4.68) point in the direction of N. Therefore, a necessary condition for xo to be in equilibrium is that the last term on the right hand-side of (4.68) also points in the normal direction, i.e. z2b2 z4 ' ^2-2' lm 2 b 2 - l + 5b2•z- = 0. (4.69) This condition is clearly satisfied when z = ± 1 and when z = ±i. Next we show that there are no other solutions to (4.69). Setting w = z2 and using 2Im(x) = x — x, (4.69) becomes '2 + 5b2w wb2 (w2 + 5b2w) wb2 • w V ; + =r^ 7 + w2 -b4 wb2 — 1 w2 — b4 159 w b 2 - l = 0. (4.70) Using the relation ww = 1, (4.70) simplifies to 4 ( 1 + t y : 1 : " 1 ' ! : ' 1 ' = o. (4.?i) v ' (b4w2 - 1) (f/ - w2) v ' It follows z = ± 1 , ±z. are the only possible equilibria. When the condition (4.69) is satisfied, we obtain to leading order from (4.68) that the equilib- rium location for a is related to D by Notice that this relation is the same for any equilibrium location. Moreover, (4.72) involves only the leading order behavior of V i ? ~ near the boundary, which is independent of the shape of the boundary. Therefore, the result (4.72) is also independent of the domain shape. Notice also that (4.72) with = n agrees with the result (4.29) derived earlier for an equilibrium spike near the boundary of a circular cylindrical domain of radius one. Next, we analyze the stability of these equilibria with respect to the direction normal to the boundary. Notice that D = D(cr) has a minimum at om = f. At this point, D(am) = Dm = \/2e2 A a 2 ^ • However, since we have assumed that a e, we have D S> Dm. Furthermore, for any given D with D » Dm, (4.72) has two solutions for a: o\ « £ and a2 S> e. By examining the sign of the right hand side of (4.68), we conclude that the equilibrium at a2 is unstable with respect to the normal direction. Alternatively, <j\ is stable with respect to the normal direction. However, our analysis is invalid for the root o~i, since o~\ <C e. Nevertheless, this formal analysis may suggest the existence of boundary spike equilibria located on the boundary of dQ. at f(z), where z = ± 1 or z = See Conjecture 4.5.1 below. We summarize: P r o p o s i t i o n 4.4.2 For an arbitrary smooth domain Q, let D be given by (4-72), where £ <C (7 C 1. Then there exists a spike equilibria location at a distance a away from the boundary. Such an equilibria is unstable in the direction normal to the boundary. 160 Next, we analyze the stability of these equilibria with respect to the tangential direction. The unit tangent vector to dfi, measured in the counterclockwise direction, is fit) = i ] m \ where z = £lt (4.73) The stability in the tangential direction is controlled by the last term on the right hand side of (4.68). The direction of this term, up to a positive constant scalar multiple, is given by _ . 1 <T( z2b2 (z 4 + 56 2 z 2 ) vit) = -\m N[*^- *-» f'(z) ( zb2 (z3 + 5b2z)\ \f'(z)\2 \ z 2 b 2 - l z*-b* (4.74a) (4.74b) where z = elt The differential equation (4.68) can be written in the form —— ~ aiiv + cov . dt (4.75) where ai and to > 0 are real. Multiplying (4.75) by f and taking real parts of the resulting expression we get R e ( x 0 T ) = w R e ( u f ) . (4.76) We decompose the velocity field as x'Q = s^N + STT, and we let 8 denote the angle between the tangential direction and v. Using the identity < a,b•>= Re(a6) for the dot product, we then reduce (4.76) to sT = UJ\V\COS6, (4.77) where cos 9 = — < T, v > , \v\ = 7 - R e \v\ zf'(z)f'(z) f z 3 + 5b2z zb2 'l/'WII/'WP v4 , rL2^2 z4 _ b4 z2b2 _ 1 = - C ( t ) I m + 5b2z- 2 4 - 6 4 2J,2 Zzb Z2b2 - 1 (4.78a) (4.78b) (4.78c) Here C(t) is some irrelevant positive scalar. 161 At the equilibrium points z = ± 1 and z = ±i we have that v points in the normal direction. Therefore, at these points, we have < T,v >= 0 and cos6 = 0. From (4.77) we observe that if d c ^ e < 0 at the equilibrium position, then we have stability in the tangential direction. At the equilibrium positions, where sin# = 1, we calculate from (4.78) that dB ~dl = C(t) Im where J~(z) = + 5b2z2 z 4 - 6 4 2b2 i2b2 At rj = 0 and t = TT/2 where z = 1 and z = i, respectively, we obtain from (4.79) that dB di = C(0)Re .F(O) d0 ~dt From (4.79) and (4.80), we calculate = - C ( 0 ) d6_ ~dl cW ~dl t=o 8b2(l + bA (b2 + l)2(b2 - l ) 2 862(1 + fr4) (b2 + 1)2(62 _ 1)2 < o , > 0. (4.79) (4.80) (4.81a) (4.81b) Since dc^d- = — j§ at the equilibrium positions, we conclude from (4.77) and (4.81) that the equilibrium position near / ( ± i ) (near / ( ± 1 ) ) is stable (unstable) in the tangential direction, respectively. We summarize our results in the following proposition. Proposition 4.4.3 For.the dumbbell-shaped domain Q = f(B), where f is given by (4.30), let a be such, that e <C a 1. For the value of D given by (4-72), there are precisely four equilibrium spike locations, all at a distance a away from the boundary. They are given by xf = f(H) — Na and x^f = / ( ± 1 ) — Na, where N is the normal to the boundary dQ. at / ( ± i ) and / ( ± 1 ) , respectively. All four equilibria are unstable in the direction N normal to the boundary. Moreover, xf (x^) are stable (unstable) in the tangential direction, respectively. In Figure 4.2 we illustrate the local vector field and stability properties of near-boundary spikes. Our results suggest that the unstable manifold of the equilibrium near-boundary spike along the x-axis connects with the stable manifold of the near-boundary spike along the y-axis in the neck region of the dumbbell. 162 Our analysis has been restricted to the case where the distance a between the spike and the boundary satisfies a S> e. We expect that the equilibrium locations of spikes that are located on the boundary of the domain should result from a competition between the zeroes of the derivative of the curvature (as for the shadow problem where D = oo) and the local behavior of the gradient of the regular part i ? m of the Green's function on the boundary. To this end, we define i?& by Rb(x) = lim Rm(x,y) + -3-ln|x - y | 4.7T (4.82) We now show that for the dumbbell-shaped domain of §4 that the minimum of Rb occurs at that point of the boundary where the curvature is at its minimum. For the dumbbell-shaped domains of §4 we obtain the following result: Proposition 4.4.4 Let O = f(B) where f is given by (4-30) and B is the unit ball. Then Ri defined in (4-82) is given by: '. 1 . / 6 4 + 26 2 cos2i + l \ i t Rb(X) = ^lnW-2b2cos2t + l ) + C > ^ X = / ( e ) ' ( 4 - 8 3 ) Here C is some constant independent of x. The proof of this result is given in Appendix §4.6. Notice that the expression inside the log term of (4.83) has its maximum at t = 0 , TT and its minimum at t — | , 3 y . Therefore, R^ has a minimum on the y axis in the neck of the dumbbell, where the curvature of 80 is at its minimum. 4.5 Discuss ion For different ranges of the inhibitor diffusivity D, we have described the bifurcation behavior of an equilibrium one-spike solution to the G M model (4.2) in a radially symmetric domain, and in a class of dumbbell-shaped domains. In a radially symmetric domain, we have calculated the bifurcation value Dc = 0(e2e2dl£), where d is the distance of the spike to the boundary, for which an equilibrium spike at the midpoint of the domain becomes stable as D decreases below Dc. For a dumbbell-shaped domain of Figure 4.1, the center at the neck also bifurcates 163 as D is increased beyond some exponentially large value; however the bifurcation occurs along the y axis, as shown in Figure 4.1(b). Since this bifurcation occurs when D is exponentially large as e —* 0, a main conclusion of our study is that spike behavior for the shadow system corresponding to D = co, has very different properties from that of spike solutions to (4.2) when D is large, but independent of e. Moreover, in §3.4, we showed that when D decreases below some 0(1) value, the spike in the neck of the dumbbell loses its stability through a pitchfork bifurcation to two stable spike locations that tend to the lobes of the dumbbell as D —> 0. Although there have been many studies of the existence and stability of boundary spikes for the shadow G M system with D = oo, the problem of constructing equilibrium boundary spike solutions for different ranges of D is largely open. The analysis in this paper has been restricted to the situation where the spike is away from the boundary, i.e. the distance cr of a spike to the boundary is such that a >̂ 0(e). Therefore, we have not described boundary spikes or spikes that are 0(e) close to the boundary. Some work on equilibrium boundary spikes for the case where D is algebraically large as e —> 0 is given in [11]. For the case where D = oo, the dynamical behavior of a boundary spike was derived in [38]. The equilibrium case was studied in [81] and [82] (see also the references therein). From these studies, it is well-known that the dynamics and equilibrium locations depend only on the curvature of the boundary of the domain. For the domain in Fig. 4.1, the boundary spike located on the y-axis is stable when D = oo. For asymptotically large values of D, we expect that the dynamics of a boundary spike depends on both the derivative of the curvature of the boundary and on the behavior of the gradient of the regular part of the Green's function Rm on the boundary. For the dumbbell-shaped domain of in Fig. 4.1, we showed in §4.sec:4.5 that this gradient vanishes at the same points where the curvature of the boundary has a local maxima or minima. This suggests the following conjecture: Conjecture 4.5.1 Suppose that 0(1) <§C D < C 0{eqec/e) for some q and c to be found. Then, a boundary spike for the domain Q, = f(B) shown in Figure 4-1 is at equilibrium if and only if its center is located on either the x or the y-axes. Furthermore, the equilibrium locations on 164 the y and the x-axes are stable and unstable, respectively. Finally, it is well-known (cf. [5], [49], [27]) that the shadow system admits unstable multi- spike equilibrium solutions where the locations of the spikes satisfy a ball-packing problem. These solutions are unstable with respect to both the large 0(1) and the exponentially small eigenvalues of the linearization. Since when D <C 1, the distance function plays a central role, the locations of the spikes should also satisfy a ball packing problem. However, in a strictly convex domain these solutions should be stable. It would be interesting to extend the analysis given here to determine the bifurcation properties, and the exchange of stability, of multi-spike solutions as D is decreased. 4.6 A p p e n d i x : T h e P r o o f of P r o p o s i t i o n 5.3 Here we prove Proposition 5.3. Let x = x(t) — f(elt), N be the normal at x, and label y(t) = x(t) - aN(t). We define h(t) by h(t) = Rm(y(t),y(t)), (4.84) and calculate h'(t) = VRm(y(t),y(t))-y'(t). (4.85) Note that x'(t) = T | / ' ( z ) | , where z — elt and T = iN is the tangential direction at x — f(z). Therefore, we have y'(t) = \f'(z)\f-acf, (4.86) where c(t) = £t\N\ is some irrelevant function. From Proposition 5.1 and using a • b = Re(ab), we obtain h\t) = I - — + 2 6 2 (z 4 + 5fc2z2) 1 47rcr 2 T T | / / ( Z ) | f\f(z)\-acT 2b2 - 1 z 4 - 6 4 + 0(<x) (4.87) 2TT l m z2b2 • (z 4 + 56 2 z 2 ) z 2 b 2 _ T_ z4_ b4 + 0(v) = —ImF(z), 165 where T is given by (4.79). Note that the singular part ^ does not enter into h'(t) since it is perpendicular to the tangent direction. Integrating (4.87), we obtain h(t) = I h'(t)dt = ±- f I m f ( z ) — . J 27T J IZ Therefore, ImF(z) 2i I 2 f 1 + bA)b2 W { w 2 ~ 1 } b2 _ - 1 i V wb2 + 1 u;62 — 1 ' w + b2 6 2 1 + w — b2 Here to = z 2 . In addition, — = 1 ^ so that H ( T ) = J . / f W 4vr J \wb2 + 1 6 2 + ATT wb2 — 1 ' w + 6 2 w — b2 (wb2 + l)(w + b2) dw, (w;62 - l)(u> - 6 2 ) + C . (4.88) (4.89a) (4.89b) (4.89c) (4.90a) (4.90b) This result can be simplified as hit) = — ln 4"7T {wb2 + l)(w + b2) (wb2 - l)(w-b2) 1 (wb2 + l)(wb2 + l)(w + b2)(w + b2) 8TT N (wb2 - l)(wb2 - l)(w - b2)(w - b2) + • 1 , ( l + 6 4 + 26 2cos2i) — — i n ^ in (1 + 6 4 - 262 cos 2t) l + 6 4 + 262 cos2?j 2+C, 47T (4.91a) (4.91b) (4.91c) (4.91d) , l + 6 4 - 262 cos2t, The constant C = C(a) depends on the distance from the boundary but not on x. Finally, for x € dQ and y € Q, we define the regular part of the boundary Green's function by S(x, y) = Rm(x, y) + — ln \x - y\. 47T (4.92) 166 Note that S is smooth and bounded for all y 6 fi. Therefore, we have Rb(x) = l im S(x, y) = l im [S(y, y) + 0(x - y)} = l im S(y, y), (4.93a) y—>x y—>x y—»x = lim/i(t) + l l n f f (4.93b) <r->0 W 47T V ' 1 , / / l + 6 4 + 26 2 cos2£ Since we are only concerned with determining points on the boundary where Rb has minima and maxima, the constant C is irrelevant. This completes the proof. • 167 Chapter 5 Q-switching instability in passively mode-locked lasers One of the applications of where pulse-like structures occur is in the field of nonlinear optics. In this chapter we consider a model arising in lasers, which exhibits localized spike-like structures. The following equations have been proposed as a continuous model of the mode-locked lasers (see [31], [41], [23]): ET = \N-1-TTW?'E + E89 (5,LA) NT = 7 A - N - NL~l / \E\2aie Jo L (5.1b) Here, 9 represents the time describing a single laser pulse and T is a slow time variable describing the modulation of its amplitude. E(9, T) represents the field satisfying the periodic boundary condition E(9,T) = E(9 + L, T), and N(T) is the inversion of population. In addition, 7 = O ( 1 0 - 4 — 10~3) is the relaxation rate of the upper level relative to the coherence lifetime. The constant A is the pump parameter defined so that A = 1 is the lasing threshold if a = 0. The saturation term in (5.1) is the same as the one used by Kartner et al [42]. This model has been studied numerically in [41]. Taking a as a bifurcation parameter, the following typical parameter values have been considered in [41]: A = 8, £ = 3800, 7 = 0.00014, 6 = 0.01. . (5.2) Using numerical simulations, it was reported in [41] that for a = 0.0025, the solution E(B) settles to a steady state in the form of a spike. However as a was increased past 0.003, fast- scale oscillations in the steady-state solution were observed: Such oscillations are called Q-switch 168 instabilities, and are undesired for most of the applications. In this chapter, we use stability analysis to predict the onset of a Q-switch instability via a Hopf bifurcation. The results of this chapter have been previously reported in [23]. From the mathematical viewpoint, we consider the following generalization of (5.1): ET = -(l + ail-iblEl^kiblE^E + NiE + Eee), (5.3a) NT = 7 (5.3b) Here, A; is a nonlinearity with k(0) = 1, p > 1, and where T is the time variable. Equation (5.1) corresponds to choosing fcW = T T t ' P = 3 ' ( 5 ' 4 ) We will also study in more detail the case of no saturation, k(t) = 1, p > 0, (5.5) for which more explicit results are possible. We scale (5.3) as follows: N = 1 + LUX, E = VA-lu (5.6a) t = uT, z =. yjuj0 (5.6b) where this yeilds, where to = - 1) (5.6c) ut = uzz(l + e2x) - u(a - x) + aB^vPkipu2)) (5.7a) xt = ^(1 - 2elX) - ^(1 + £2x)j J u2 (5.7b) a = a—, 0 = b(A — l), I = \fuih, £1 = 777, e2 = u>. (5.8a) 169 The typical values (5.2) then become 8 = 0.07, £ l = 0.0032, e2 = u = 0.044, I = 800. (5.9) The numerical result from [41] is that for k and p as given in (5.4), there is a Hopf bifurcation as a passes through ac = 0.068. We will treat £i,£2 as small parameters in this problem. Our goal is to study the asymptotic stability in terms of these parameters. 5.1 S tab i l i ty analysis Let u = u*,x = x* be the steady state solution. It is convenient to further rescale the space variable as follows: z = dz, u = hz, x = x (5.10a) where d 2 = a~X* , . h = (a-x*)^Ta-^0-^. (5.10b) 1 + e2x* Dropping the bars, (5.7) becomes ut = uzz(l + £2x)^~——— - u(a - x) + (a - x*)upk(cu2), (5.11a) 1 + £2x* 1 1 h? fld xt = -(l-2e1x)--(l + e2x)— u2 (5.11b) o 2 V L ' 2 V ' * 'Id where 2 2 _ c=(a — x*)p~1a p-1. (5.11c) The steady-state equations therefore become 0 = u*zz-u* + u*pk(cu*2) (5.12a) 170 Next we examine the stability. Letting u = u* + eXt(p(z), x = x* + eMib(z we obtain, up to 0 ( £ i , £ 2 ) , the following eigenvalue problem: A ,„ d a — x* 2 rid W = -ert - \^fA - (1 + c 2 x - ) £ ^ Using (5.15) to solve for ib and substituting it into (5.15) we obtain + ^zz£2 t>u a — x* -<f> = LQ4> u a — x ' 1 * + u„e2 1 + e 2x* h? A + £1 + \e2 Id J0 id where For convenience we rescale Lot =f r-<f>+J-;(u*pk(cu*2) A = a — x* Dropping bars we obtain u r , , * , n , x > 2 fld , * de/ 1 + £ 2 X * ' a J o (a - x*) ((a - x*)A + £1 + i £ 2 ) Since this equation is linear in <fr, we may scale it to obtain 1 _ h__ rld x l d Jo (LQ - \)<f> = u* + 0(e2) 4>u* We now look for a hopf bifurcation. That is assume that A is purely imaginary. We let A = i A j , <j> = 4>R + i<f>i. to obtain £o<^fl + A/(/)/= u* + 0 ( £ 2 ) , Lo<j)i - \i<j)R = 0 This equation decouples: + \]4>r = A ,u* + 0 ( £ 2 ) , Lg^fl + Affo = Lou* (5.13) (5.14) (5.15) (5.16) (5.17) (5.18) (5.19) (5.20a) (5.20b) (5.21) (5.22) (5.23) 171 and hence (5.20a) becomes i ^ ^ l ± M = 2 ^ > H « M R ( A , ) (5.24b) Together, equations (5.24), (5.12) and (5.11c) yield a system of three equations for three un- knowns Xj, x* and a which determine the hopf bifurcation. However the solution of this system requires extensive computations since 4>J,4>R have to be determined numerically for any given a, Xj as a solution to the boundary value problem of fourth order. In practice, for reasonable nonlinearity k such as of type (5.4) or (5.5), we may assume that c = 0(1), or equivalently a-x* = 0(a). (5.25) It then follows that d = 0(y/a), h~2 = O(0). Furthermore, from (5.12b) we obtain / u*2 = 0(iPy/a), (5.26) Jo from where we conclude that dl = 0(l^/a) >̂ 1, since otherwise JQdl u*2 — 0{ly/a), which contradicts (5.26) and the fact that 0 <C 1. Therefore, rdl u* ~ w, J u*2 ~ j w2, (5.27) where w satisfies the same O D E (5.12a) as it*, but on the whole space: 0 = wzz -w + wpk(cw2), (5.28a) w(z) = ae~W as \z\ -+ oo (5.28b) Furthermore ldh~2 = O(ly/a0) and J w2 = 0(1) and hence it follows from (5.27) and (5.26) that ldh~2 = 0(1). Thus the left hand side of (5.24b) is of 0(a.£i), and therefore small. On the other hand expanding (j>R, <pj as a series in JJ and using (5.22) we obtain I f 1 Wi(A/) = J w L ° w + ° ^ ) ' (5'29a) M{-Xl) = hS w2+°{hI)- (5-29b) 172 Since the left hand side of (5.24b) is small, we may assume that <J>R is small and therefore the expansion (5.29a) is valid. Thus Xj is big and therefore the expansion (5.29b) is also valid. Substituting (5.29) into (5.24) and eliminating A/ , then using the approximation (5.27) we obtain £ i + 5 £ 2 _ / WLQW a — x* J w2 From (5.12b) and (5.11c) we obtain (5.30) Combining this with (5.30) we obtain an equation involving only c: (l(3)2(e1 + l-e2) = c 2 w2^j wLow^j (5.32a) yL2b2A(A - \ ) 2 = c2(^J w2^j wL0v?j .. (5.32b) Once c is so determined, we may finally determine a from (5.31),(5.32a) and (5.11c): a = (pl)~2 ^J°° w2^j (5.33a) p — 5 _j_2 p —5 = (l0)^-2(e1 + l-e2)-^ (^j w2^ 4 ^ ™ L o W ) 4 (5.33b) Using (5.8) we finally obtain a = ^(A-l)^EA5^E(Lb)^F(c)p) (5.34a) where p+3 ' p - 5 F(c,p) = ^ w2^j 4 ^ tuLow) 4 (5.34b) Thus we obtain the following characterisation of the Hopf bifurcation of (5.7): Proposition 5.1.1 Let w be the entire solution to (5.28) where c satisfies (5.32b), with c = O(l). Then there is a Hopf bifurcation that occurs for the system (5.3) for the value of a given by (5.34). 173 5.2 H o p f bifurcat ion for for p = 3, k(t) = T T T To find c of Proposition 5.1.1 we need to be able to evaluate w2 and WLQW for various values of c. There are two ways of doing this efficiently. Note that (5.28) admits a first integral /2 2 w w ~2 2 rw + f(w,c) = 0 where /(w,c) = / tpk(ct2)dt (5.35) /o Thus = w2 , d w and so V™ 2 -2/Kc) 2 _ n 2 du; /• T „ r . dto wz = 2 wA wL0w = 2 WLQW , (5.36) Jo- ^w2-2f(w,c) J Jo ^w2-2f{w,c) ^ ' where wm is the maximum of w given by wm2-2f(wm,c) = 0. (5.37) However, the integral on the right hand side of (5.36) is singular at the right endpoint and a numerical approximation on a uniform grid using for example Simpson's method gives a very bad approximation. This can be overcome by developing this integral in Taylor series at the singular endpoint. A simpler and a numerically cheap approach is to instead integrate the initial value problem w(0) = wm, A(0) = 0, A' = 2w2, w" -w + wpk(cw2) = 0, where wm and c are related by (5.37), using a standard Runga-Kutta method from zero until t\ where say w(tx) < 10~ 8 (in typical examples we tried, ti was about 10-15). Then J^w2 is numerically approximated by A(t{). The integral WLQW is analogously evaluated. A physically relevant case studied in [41] is p = 3, k(t) = y ^ . Then we compute: 2w3 Low = K (1 + cw2)2 and w2 - 2/(«;, c) = ^ 1 - ^ w2 + -\\n{l + cw2). (5.38) For parameter values (5.2) used in [41], we used the method outlined above to numerically obtain c = 0.580 (which is of O(l) ) , wm = 2.515, f^w2 = 18.78, J^wLow = 12.60 which 174 yields the predicted bifurcation value of a = 0.065 or a = 0.0029. This agrees to all significant digits with the value of ac = 0.003 reported in [41], as observed from a full numerical simulation of (5.3). Figure 5.1 lists the threhold value a at which the hopf bifurcation occurs, as a function of b. 0.02 0.018 0.02 Figure 5.1: The hopf bifurcation diagram in the b — a plane. The solid curve is the numerically computed bifurcation threhold from Proposition 5.1.1. Instability occurs above the solid curve. The dashed curve is the asymptotic approximation valid for small 6, given by (5.46a). 5.3 H o p f bifurcat ion wi thou t sa tura t ion Next, we examine the case k = 1 for arbitrary p. Equations (5.37) and (5.36) then become p+ 1\ p-i 2 0 / 7 1 dw o / p + l V " 1 f1 v d v w = 2 w—====== = 21 1 ' 1 - 2 P+i o \ / l - VP' 1 To evaluate J wLow, note that Low = (p — l)wp. From (5.35) we have 2 j W'2 - j W2 + -Ay- J WP+1 = 0. (5.39) (5.40) (5.41) Multiplying (5.28a) by w, integrating, and using integration by parts on the first term we obtain - J w'2 - J w2 + J wp+1 = 0. (5.42) 175 Thus we obtain WLQW = =KP+'>(;-!> / > . (5.43) For the cases p = 2, 3, 5, the integral (5.40) is easily evaluated using elementary techniques. More generally, we have / ' " - r ^ ) r ( ^ > ( 5 4 4 ) In particular it is possible to to evaluate this integrals in terms of elementary functions when the argument to F is of the form n/2 with n an integer. This happens when p is of the form 2 ± m Q r 4+m w ^ & n j n t e g e r m m 0 We summarize these results below. Proposition 5.3.1 Suppose that k = 1 in (5.3). Then a Hopf bifurcation occurs in (5.3) for the value of a given by a = (1A)^ ((A-VjLb)1^ F(p), (5.45a) where 2-1 F(p) = 2 ^ ( P + 1 ) ^ ( P - 1 ) =5*(p + 3 ) ^ ( r ( f ) 2 • ' ( 5 ' 4 5 b ) For special values of p we have the following: " = 3 : '=j£m-  F=v~6 •  (546a) p = 5 : ""^p^W F = i " 2 = 7 4 0 2 <5-46b) ; 2 p = ^ : a = 151 ( 4 ( / ^ ) L 6 ) 3 - ^ = ^ 1 5 ^ = 2.099 (5.46c) p = - : a = 2 .89(7i4) i i [ (A- l )Lfe]5 , F = —5~542* = 2.89 (5.46d) p = 2 : a = 2.136(7A>s(( i4- l)L6)~5, F = 5*6~* = 2.136 . , (5.46e) When p = 5, £/ie uaZue o/ a a£ which the Hopf bifurcation occurs does not depend on 7 . When p — | , i£ does nof; depend on A if A is large. 176 5.4 Discuss ion A n open problem is to study how Hopf bifurcation behaves as we change the saturation pa- rameter b. For very small b, which corresponds to the limit of no saturation, the asymptotic expression for the Hopf bifurcation was derived in (5.46a). It would be also interesting to find an asymptotic expression for large 6. This corresponds to the case c near 1 in (5.38). Setting c = 1 — 6, where 0 < 5 <C 1, we then obtain g(w) = w2 - 2 / ( i u , c) 5w2 + (1 + 28) l n ( l + w2). Then setting j = 0 w e obtain In w2 W; 2 ' WM » 1. This gives an expression for the height of the spike. To compute the hopf bifurcation in this case, we need to evaluate the integrals in (5.36) asymptotically. It is an open question to complete this computation. u(l/2) 18- 16 14; 12: 10- ' J -400 -200 200 400 -- /1 / A A A A 133 610/ 620: I 630. 640. 630. (a) (b) Figure 5.2: (a) Solution profile u(z) at different times, (b) Height of the spike and x(t) vs. time. Here, a = 0.1,0 = 0.07,/ = 800,p = 3,k(t) = ^-t,e1 = 0.0032 and e2 = 0.044. 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Mountain View 2 0
Beijing 2 1
Ashburn 2 0
Nottingham 1 0
Sunnyvale 1 0
Stockholm 1 0
Gandhinagar 1 0
Rochester 1 0

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