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Behaviour of solutions to the nonlinear Schrödinger equation in the presence of a resonance Coles, Matthew Preston 2017

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Behaviour of Solutions to theNonlinear Schro¨dinger Equation in thePresence of a ResonancebyMatthew Preston ColesB.Sc., McMaster University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Matthew Preston Coles 2017AbstractThe present thesis is split in two parts. The first deals with the focusing Non-linear Schro¨dinger Equation in one dimension with pure-power nonlinearitynear cubic. We consider the spectrum of the linearized operator about thesoliton solution. When the nonlinearity is exactly cubic, the linearized oper-ator has resonances at the edges of the essential spectrum. We establish thedegenerate bifurcation of these resonances to eigenvalues as the nonlinearitydeviates from cubic. The leading-order expression for these eigenvalues isconsistent with previous numerical computations.The second considers the perturbed energy critical focusing NonlinearSchro¨dinger Equation in three dimensions. We construct solitary wave solu-tions for focusing subcritical perturbations as well as defocusing supercriticalperturbations. The construction relies on the resolvent expansion, which issingular due to the presence of a resonance. Specializing to pure power fo-cusing subcritical perturbations we demonstrate, via variational arguments,and for a certain range of powers, the existence of a ground state soliton,which is then shown to be the previously constructed solution. Finally,we present a dynamical theorem which characterizes the fate of radially-symmetric solutions whose initial data are below the action of the groundstate. Such solutions will either scatter or blow-up in finite time dependingon the sign of a certain function of their initial data.iiLay SummaryWe conduct a mathematically motivated study to understand qualitativeaspects of the nonlinear Schro¨dinger equation. For this summary, however,we imagine our equation as describing the positions of many cold quantumparticles in a cloud. A group of particles may cluster together and persistin this configuration for all time; this structure being called a soliton. Oneaspect we are interested in is the stability of the soliton. It may be stable - asmall disruption of the system will be weathered and the soliton will remain,or unstable - a perturbation will cause the particles to break up, destroyingthe soliton. The soliton also impacts the overall dynamics of the equation.If the particles do not have enough mass or energy to form a soliton, theyspread out and scatter. On the other hand, with too much mass or energy,they may blow-up, coming together to form a singularity.iiiPrefaceA version of Chapter 2 has been published in [19]. I conducted much of theanalysis, all of the numerics, and wrote most of the manuscript.A version of Chapter 3 has, at the time of this writing, been submittedfor publication to an academic journal. The preprint is available here [20].I conducted much of the analysis and wrote most of the manuscript.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Nonlinear Schro¨dinger Equation . . . . . . . . . . . . . 11.2 Conserved Quantities, Scaling, and Criticality . . . . . . . . 21.3 Scattering and Blow-Up . . . . . . . . . . . . . . . . . . . . . 31.4 Solitary Waves and Stability . . . . . . . . . . . . . . . . . . 51.5 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 The 1D Linearized NLS . . . . . . . . . . . . . . . . . . . . . 81.7 A Perturbation of the 3D Energy Critical NLS . . . . . . . . 102 A Degenerate Edge Bifurcation in the 1D Linearized NLS 182.1 Setup of the Birman-Schwinger Problem . . . . . . . . . . . 182.2 The Perturbed and Unperturbed Operators . . . . . . . . . . 222.3 Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . 272.4 Comments on the Computations . . . . . . . . . . . . . . . . 36vTable of Contents3 Perturbations of the 3D Energy Critical NLS . . . . . . . . 403.1 Construction of Solitary Wave Profiles . . . . . . . . . . . . 403.1.1 Mathematical Setup . . . . . . . . . . . . . . . . . . . 403.1.2 Resolvent Estimates . . . . . . . . . . . . . . . . . . . 413.1.3 Solving for the Frequency . . . . . . . . . . . . . . . . 483.1.4 Solving for the Correction . . . . . . . . . . . . . . . 563.2 Variational Characterization . . . . . . . . . . . . . . . . . 653.3 Dynamics Below the Ground States . . . . . . . . . . . . . 784 Directions for Future Study . . . . . . . . . . . . . . . . . . . 874.1 Improvements and Extensions of the Current Work . . . . . 874.2 Small Solutions to the Gross-Pitaevskii Equation . . . . . . . 88Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93viList of Tables2.1 Numerical values for 8α2. . . . . . . . . . . . . . . . . . . . . . 39viiList of Figures2.1 The two components of Pw1. . . . . . . . . . . . . . . . . . . . 382.2 The two components of function g. . . . . . . . . . . . . . . . . 39viiiAcknowledgementsI am grateful to my supervisors, Stephen Gustafson and Tai-Peng Tsai,for suggesting interesting problems, providing stimulating discussion, andsupplying constant mentorship. For keeping me motivated, I thank myfellow graduate students. I am in debt to friends and family, in particularmy parents, for their enduring emotional support. I acknowledge partialfinancial support from the NSERC Canada Graduate Scholarship.ixfor DainaxChapter 1Introduction1.1 The Nonlinear Schro¨dinger EquationThe nonlinear Schro¨dinger equation (NLS) is the partial differential equation(PDE)i∂tu = −∆u± |u|p−1u. (1.1)It attracts interest from both pure and applied mathematicians, with ap-plications including quantum mechanics, water waves, and optics [35], [74].Before discussing the applications we establish some terminology and no-tation. Here u = u(x, t) is a complex-valued function, x is often thoughtof a spacial variable (in n real dimensions, x ∈ Rn), and t is typically thetemporal variable (t ∈ R). The operator ∂t is the partial derivative in timeand ∆ = ∂2x1 + . . . + ∂2xn is the laplacian. We have chosen the pure powernonlinearity, ±|u|p−1u, with power p ∈ (1,∞), but could also replace thisnonlinearity with a more complicated function of u. The nonlinearity withthe negative sign, −|u|p−1u, will be referred to as the focusing or attractivecase while the positive, +|u|p−1u, is the defocusing or repulsive nonlinearity.In applications the most common nonlinearities are the cubic (p = 3) andthe quintic (p = 5). Also common is the cubic-quintic nonlinearity: the sumor difference of cubic and quintic terms.Let us now briefly describe some of the contexts in which the NLS canbe applied. Firstly, in quantum physics the nonlinear Schro¨dinger equation,and the closely related Gross-Pitaevskii equation (which is NLS subject toan external potential), describes the so called Bose-Einstein condensate. ABose-Einstein condensate is a cloud of cold bosonic particles all of which arein their lowest energy configuration. In this way, the mass of particles can bedescribed by one wave function; that is u(x, t). In this context the absolutevalue |u(x, t)|2 describes the probability density to find particles within thecloud at point x in space, and at time t. The inter-particle forces may beeither attractive or repulsive, which, corresponds to focusing or defocusingnonlinearities, respectively. See [74] or the review [14] for more information.11.2. Conserved Quantities, Scaling, and CriticalityThe NLS has also applications in water waves, a pursuit that dates backto [91]. For a thorough description see, for example, Chapter 11 of [74](and references therein). The water-wave problem concerns the dynamicsof a wave train propagating at the surface of a liquid. In deep water, thesolution u(x, t) to the 1D cubic (p = 3) NLS describes an envelope whichcaptures the behaviour of a wave which is modulated only in the directionin which it propagates.In nonlinear optics, the function u describes a wave propagating in aweakly nonlinear dielectric (see again [74], Chapter 1). In this case ∆ is theLaplacian transverse to the propagation and the time variable t is replacedby the spacial variable along the direction of propagation. For example in3D where x = (x1, x2, x3) the NLS may take the formi∂x3u = −(∂2x1 + ∂2x2)u± |u|2uwhere x3 is in the direction of propagation.On the pure mathematical side, the NLS is interesting as a general modelof dispersive and nonlinear wave phenomena. The nonlinear Schro¨dingerequation provides an arena to develop techniques which can be applied toother nonlinear dispersive equations. The NLS is often technically simplerthan other equations [15], such as the Korteweg-de Vries equation (KdV),the nonlinear wave equation (NLW), Zakharov system, Boussinesq equation,and various other water-wave models.1.2 Conserved Quantities, Scaling, and CriticalityThe facts we review here are standard and can be found in, for example, thebooks [15, 35, 74, 77].The NLS (1.1) has the following conserved quantitiesM(u) = 12∫Rn|u|2 dx, E(u) =∫Rn12|∇u|2 ± 1p+ 1|u|p+1 dx (1.2)often called mass and energy, respectively. These quantities are conservedin time, that is: M(u(x, t)) = M(u(x, 0)) and E(u(x, t)) = E(u(x, 0)) forsufficiently smooth solutions.The scalingu(x, t) 7→ uλ(x, t) := λ−2/(p−1)u(λ−1x, λ−2t) (1.3)preserves the equation. That is if u(x, t) is a solution to (1.1) then uλ(x, t)is also a solution.21.3. Scattering and Blow-UpWhen the power p and dimension n are chosen according to the followingrelationp =4n+ 1our scaling (1.3) also preserves the mass, that isM(uλ(·, t)) =M(u(·, λ−2t)).Such an equation is called mass critical. For example, in one dimension(n = 1) the quintic (p = 5) NLS is mass critical. For values of p < 4/n+ 1we call the equation mass sub-critical and for p > 4/n + 1 we say masssuper-critical.Similarly, ifp = 1 +4n− 2 , n ≥ 3then (1.3) preserves the energy, so E(uλ(·, t)) = E(u(·, λ−2t)). We call thisequation an energy critical equation. For example, in three dimensions(n = 3) the quintic (p = 5) NLS is energy critical. Again we refer top < 1 + 4/(n − 2) as energy sub-critical and p > 1 + 4/(n − 2) as energysuper-critical. Note that for dimensions one and two all equations withp <∞ are energy sub-critical.1.3 Scattering and Blow-UpThe overall dynamics of the equation are affected by the power p’s relationto the mass and energy critical values. By overall dynamics we mean thelong time behaviour of the solution subject to an initial condition, ie. theCauchy problem {i∂tu = −∆u± |u|p−1uu(x, 0) = u0(x). (1.4)We seek theorems which characterize the solution’s eventual behaviour fora large class of initial conditions u0.One possibility is that the solution will scatter (Chapter 7 of [15], Chap-ter 3.3 of [74]). This means that the solution u(x, t) will eventually look likea solution to the linear Schro¨dinger equation. That is, after sufficient time,all nonlinear behaviour has disappeared and only linear behaviour remains.Solutions to the linear Schro¨dinger equation (Chapter 2 of [15], Chapter3.1 of [74]) {i∂tu = −∆uu(x, 0) = u031.3. Scattering and Blow-Upevolve in a predictable way. The mass, or L2 norm, is preserved while theL∞ norm (supremum) decays according to‖u(x, t)‖L∞(Rn) . t−n/2‖u0‖L1(Rn). (1.5)The above norms are defined as‖u(x, t)‖L∞(Rn) = supx∈Rn|u(x, t)| and ‖u(x, t)‖L1(Rn) =∫Rn|u(x, t)| dxand when we write f(t) . g(t) we mean that there is a constant C, inde-pendent of t, such that f(t) ≤ Cg(t).We may think of the linear Schro¨dinger equation as modeling a freequantum particle. The particle has a tendency to “spread out” due tomomentum uncertainty, even if it is well localized to start.A scattering theorem will then have the following form: for all u0 ∈ X(or some subset of X) we have‖u(x, t)− ulin‖X → 0, as t→∞.Here ulin is a solution to the linear Schro¨dinger equation (chosen accordingto the particular u0), X is an appropriate function space, and ‖ · ‖X is anorm on this space.Scattering is, to some extent, expected in the defocusing (repulsive) case.We think about the Laplacian as being a force for dispersion, as evidencedby the behaviour of the linear Schro¨dinger equation, and we think of thedefocusing nonlinearity as a repellent to the gathering of mass. We singleout the result [38], a scattering theorem in the energy space X = H1 fordefocusing NLS with power p between mass critical and energy critical,1 +4n< p < 1 +4n− 2 ,in dimensions n ≥ 3. The space H1 consists of functions u whose H1 norm,‖u(x, t)‖2H1(Rn) = ‖u(x, t)‖2L2(Rn) + ‖u(x, t)‖2H˙1(Rn)=∫Rn|u(x, t)|2 dx+∫Rn|∇u(x, t)|2 dx,is finite.For the focusing equation, with n ≥ 3, X = H1, 1 + 4/n < p < 1 +4/(n − 2), we have the scattering theory [73]. The key difference now inthe focusing case is the assumption that the energy (H1) norm of the initial41.4. Solitary Waves and Stabilitycondition, u0, be small. Here we see that if we do not have enough mass orenergy to begin with, the force of dispersion will win out over the focusingnonlinearity’s tendency to attract mass together.In the focusing equation, however, there are other possible fates for so-lutions besides scattering. If too much mass is assembled, the attractivenonlinearity may cause the solution to break down in finite time; that is,some norm of the solution will blow-up. For example, (Chapter 5 of [74])consider the focusing NLS with p ≥ 1 + 4/n. There exist initial conditionsin u0 ∈ H1 such that there exists a time t∗ <∞ such thatlimt→t∗‖∇u(x, t)‖L2 =∞.Important for the discussion of mass super-critical and energy sub-criticalequations above, scattering in the defocusing case and blow-up in the focus-ing case, is the local existence theory. The H1 local theory ensures an initialdata u0 ∈ H1 will generate a solution u(x, t) with continuous in time H1norm up to some time T . Here T is a non-increasing function of ‖u0‖H1 .Repeatedly applying the local theory together with an a priori H1 bound(such as in the defocusing case) then yields global existence, ie. the solutionexists for all time (the maximal time of existence Tmax =∞). In the absenceof an a priori H1 bound (focusing case) it’s possible that the maximal timeof existence for our initial data is finite. Indeed, the local theory provides thefollowing blow-up criterion: if Tmax <∞ then limt→T−max ‖u(·, t)‖H1 =∞.The energy critical case, p = 1 + 4/(n − 2), is more challenging sincethe blow-up criterion generated by the corresponding local theory is morecomplicated. Nevertheless, in the defocusing energy critical case the globalexistence and scattering theory is more or less complete. First we havescattering theorems [10] [11] in dimensions n = 2, 3, also [40] in dimensionn = 3, and then [76] in dimensions n ≥ 5, all where the initial data u0 wasassumed to be radial. The radial assumption was removed in [21] for n = 3and later in [67] for n = 4 and finally in [83] for n ≥ 5. Scattering in thedefocusing energy super-critical case remains a substantial open problem.We discuss scattering and blow-up for the focusing energy critical equa-tion (a study which was initiated in [54]) in Section 1.7 and Chapter 3.1.4 Solitary Waves and StabilityIn the focusing NLS,i∂tu = −∆u− |u|p−1u,51.4. Solitary Waves and Stabilitythe dispersive force of the Laplacian may be balanced by the attractivenonlinearity leading to solutions that neither scatter nor blow-up. Our NLSmay admit solutions of the formu(x, t) = eiωtQ(x) (1.6)often called solitary wave solutions or solitons. We think of solitary wavesas bound states or equilibrium solutions: the time dependence is confinedto the phase and the absolute value, |u|, is preserved. We further classifysolitons as ground states if they minimize the actionSω = E + ωMamong all non-zero solutions of the form (1.6).The time independent function Q(x) is the solitary wave profile, which,satisfies the following elliptic partial differential equation−∆Q− |Q|p−1Q+ ωQ = 0. (1.7)The above elliptic equation (1.7) is well studied, with results going backto [72] and [9]. Much of the present thesis concerns solitary waves. Inparticular, their existence, stability, and impact on the overall dynamics ofthe equation.Ground state solitary waves are stable for mass sub-critical powers of p[16] and unstable for mass critical [88] and mass super-critical powers [7]. Wecall solitary waves (orbitally) stable if initial conditions close to the solitonproduce solutions which remain close to the soliton (modulo symmetries ofspatial translation and phase rotation) for all time. More precisely, let ϕ(x)be the spatial profile of the ground state, u0(x) ∈ H1 an initial condition,and u(x, t) the solution generated by the initial condition. We say eiωtϕis (orbitally) stable if for every ε > 0 there exits a δ(ε) > 0 such that if‖φ− u0‖H1 ≤ δ(ε) thensupt∈Rinfθ∈Rinfy∈Rn‖u(·, t)− eiθφ(· − y)‖H1 < ε.Otheriwise, we say a soliton is unstable. In fact, both [88] and [7] demon-strate that some initial condition close to the soliton blows up in finite time.To study stability one can employ variational methods, as in [16], [88], [7]or else study the linearized operator around the soliton (see [39], [41], [89]).Understanding the spectrum of the linearized operator is often necessary toestablish asymptotic stability results such as those obtained in [6, 13, 22,61.5. Resonance23, 26, 36, 65, 68]. A soliton is asymptotically stable if initial conditionsclose to the soliton produce solutions which converge to a (nearby) solitonas t→∞. The study we initiate in Section 1.6 and Chapter 2 concerns thelinearized operator about 1D solitons.1.5 ResonanceBy resonance, or resonance eigenvalue, we mean a ‘would be’ eigenvalue,usually at the edge of the continuous spectrum. The resonance is not a trueeigenvalue because its resonance eigenfunction does not have sufficient decayto be square integrable. The appearance of a resonance, in the linearizedoperator about a soliton for example, is a non-generic occurrence. While itis non-generic, however, it does appear in a few key equations; in particularthose NLS that we study in Chapter 2 and Chapter 3. Such a resonance maycomplicate analysis, for example by making singular the resolvent expansionand slowing the time-decay of perturbations to a soliton.For example, let us consider the linear Schro¨dinger operatorH := −∆ + V (1.8)in n dimensions where V = V (x) is a potential. We may have a resonanceeigenvalue λ with resonance eigenfunction ξ such thatHξ = λξbut ξ /∈ L2(Rn).In 3D we require the resonance ξ be in L3w(R3), the weak L3 space.Therefore, ξ may decay like 1/|x| at infinity. The original paper [47] com-putes several terms in the resolvent expansion in the presence of a resonancein 3D. The resolvent is singular as λ→ 0 and takes the following form (as-suming we have no edge-eigenvalue):(H + λ2)−1 = O(1λ)(1.9)as an operator on suitable spaces. Moreover, the time-decay estimate (1.5)is retarded and decays in time like t−1/2 instead of t−3/2. A restated versionof these results are crucial in the analysis of Chapter 3.In 1D a resonance ξ may not decay at all. If Hξ = λξ and ξ /∈ Lpfor any p < ∞ but ξ ∈ L∞ then we regard ξ as a resonance. The morerecent work [48] provides a unified approach to resolvent expansions across71.6. The 1D Linearized NLSall dimensions, and so in 1D in particular, which, we rely on in Chapter 2.The resolvent expansion itself in 1D appears similar to (1.9).Interestingly, resonances in the Schro¨dinger operator only appear in di-mensions 1-4 [46–48] not in dimensions n ≥ 5 [45].1.6 The 1D Linearized NLSConsider now the following focusing NLS in 1 space dimensioni∂tu = −∂2xu− |u|p−1u. (NLSp)Chapter 2 deals with the above equation and so we supply here some back-ground, motivation, and connection to previous works.The above (NLSp) is known to exhibit solitary waves. Indeed, since weare in 1D the solitons are available in the following explicit formu(x, t) = Qp(x)eitwhereQp−1p (x) =(p+ 12)sech2(p− 12x).One naturally asks about the stability of these waves, which leads immedi-ately to an investigation of the spectrum of the linearized operator governingthe dynamics close to the solitary wave solution.The linearized operator is obtained by considering a perturbation of thesolitary wave,u(x, t) = (Qp(x) + h(x, t))eit,and neglecting all but the leading order in the resulting system. After wecomplexify, ie. letting ~h =(h h¯)T, we obtain the linear systemi∂t~h = Lp~hwhereLp =(1 00 −1)((−∂2x + 1 00 −∂2x + 1)− 12(p+ 1 p− 1p− 1 p+ 1)Qp−1p)is the linearized operator. See section 2.1. Systematic spectral analysis ofthe linearized operator has a long history (eg. [39, 89], and for more recentstudies [17, 30, 85, 86]).81.6. The 1D Linearized NLSThe principle motivation for Chapter 2 comes from [17] where resonanceeigenvalues (with explicit resonance eigenfunctions) were observed to sit atthe edges (or thresholds) of the spectrum for the 1D linearized NLS problemwith focusing cubic nonlinearity. Numerically, it was observed that the sameproblem with power nonlinearity close to p = 3 (on both sides) has a trueeigenvalue close to the threshold. We establish analytically the observedqualitative behaviour. Stated roughly, the main result of Chapter 2 is:for p ≈ 3, p 6= 3, the linearization of the 1D (NLSp) about its soliton haspurely imaginary eigenvalues, bifurcating from resonances at the edges of theessential spectrum of linearized (NLS3), whose distance from the thresholdsis of order (p− 3)4.The exact statement is given as Theorem 2.3.1 in Section 2.3, and includesthe precise leading order behaviour of the eigenvalues.The eigenvalues obtained here, being on the imaginary axis, correspondto stable behaviour at the linear level. A further motivation for obtainingdetailed information about the spectra of linearized operators is that suchinformation is a key ingredient in studying the asymptotic stability of solitarywaves: see [6, 13, 22, 23, 26, 36, 65, 68] for some results of this type. Suchresults typically assume the absence of threshold eigenvalues or resonances.The presence of a resonance is an exceptional case which complicates thestability analysis by retarding the time-decay of perturbations. Nevertheless,the asymptotic stability of solitons in the 1D cubic focusing NLS was recentlyproved in [29]. The proof relies on integrable systems technology and so isonly available for the cubic equation. The solitons are known to be stable inthe (weaker) orbital sense for all p < 5 (the so-called mass subcritical range)while for p ≥ 5 they are unstable [41, 90], but the question of asymptoticstability for p < 5 and p 6= 3 seems to be open. The existence (and location)of eigenvalues on the imaginary axis, which is shown here, should play a rolein any attempt on this problem.The generic bifurcation of resonances and eigenvalues from the edge ofthe essential spectrum was studied by [28] and [84] in three dimensions.Edge bifurcations have also been studied in one dimensional systems usingthe Evans function in [52] and [53] as well as in the earlier works [50], [51]and [64]. We do not follow that route, but rather adopt the approach of[28, 84] (going back also to [48], and in turn to the classical work [47]),using a Birman-Schwinger formulation, resolvent expansion, and Lyapunov-Schmidt reduction.Our work is distinct from [28, 84] due to the unique challenges of working91.7. A Perturbation of the 3D Energy Critical NLSin one dimension, in particular the strong singularity of the free resolventat zero energy, which among other things increases by one the dimensionof the range of the projection involved in the Lyapunov-Schmidt reductionprocedure.Moreover, our work is distinct from all of [28, 52, 53, 84] in that we studythe particular (and as it turns out non-generic) resonance and perturbationcorresponding to the near-cubic pure-power NLS problem. Generically, aresonance is associated with the birth or death of an eigenvalue, and such isthe picture obtained in [28, 52, 53, 84]: an eigenvalue approaches the essen-tial spectrum, becomes a resonance on the threshold and then disappears.In our setting, the eigenvalue approaches the essential spectrum, sits on thethreshold as a resonance, then returns as an eigenvalue. The bifurcationis degenerate in the sense that the expansion of the eigenvalue begins athigher order, and the analysis we develop to locate this eigenvalue is thusconsiderably more delicate.1.7 A Perturbation of the 3D Energy CriticalNLSIn Chapter 3 we consider Nonlinear Schro¨dinger equations in three spacedimensions, of the formi∂tu = −∆u− |u|4u− εg(|u|2)u, (1.10)where ε is a small, real parameter. Equation (1.10) is a perturbed versionof the focusing energy critical NLS. This section is devoted to introducingthe above equation, providing some background on the unperturbed criticalequation, and stating the main theorems of Chapter 3.The mass and energy of (1.10) areM(u) = 12∫R3|u|2 dx, Eε(u) =∫R3{12|∇u|2 − 16|u|6 − ε2G(|u|2)}dxwhereG′ = g. We are particularly interested in the existence (and dynamicalimplications) of solitary wave solutions of the formu(x, t) = Q(x)eiωtof (1.10). We will consider only real-valued solitary wave profiles, Q(x) ∈ R,for which the corresponding stationary problem is−∆Q−Q5 − εf(Q) + ωQ = 0, f(Q) = g(Q2)Q. (1.11)101.7. A Perturbation of the 3D Energy Critical NLSSince the perturbed solitary wave equation (1.11) is the Euler-Lagrangeequation for the actionSε,ω(u) := Eε(u) + ωM(u) ,the standard Pohozaev relations [34] give necessary conditions for existenceof finite-action solutions of (1.11):0 = Kε(u) := ddµSε,ω(Tµu)∣∣∣∣µ=1=∫|∇Q|2 −∫Q6 + ε∫ (3F (Q)− 32Qf(Q)))0 = K(0)ε,ω(u) :=ddµSε,ω(Sµu)∣∣∣∣µ=1= ε∫ (3F (Q)− 12Qf(Q))− ω∫Q2(1.12)where(Tµu)(x) := µ32u(µx), (Sµu)(x) := µ12u(µx)are the scaling operators preserving, respectively, the L2 norm and the L6(and H˙1) norm, and F ′ = f (so F (Q) = 12G(Q2)).The corresponding unperturbed (ε = 0) problem, the 3D quintic equa-tioni∂tu = −∆u− |u|4u, (1.13)is energy critical ie. the scalingu(x, t) 7→ uλ(x, t) := λ1/2u(λx, λ2t)which preserves (1.13), also leaves invariant its energyE0(u) =∫R3{12|∇u|2 − 16|u|6}dx, E0(uλ(·, t)) = E0(u(·, λ2t)).One implication of energy criticality is that (1.13) fails to admit solitarywaves with ω 6= 0 – as can be seen from (1.12) – but instead admits theAubin-Talenti static solutionW (x) =(1 +|x|23)−1/2, ∆W +W 5 = 0, (1.14)whose slow spatial decay means it fails to lie in L2(R3), though it does fallin the energy spaceW 6∈ L2(R3), W ∈ H˙1(R3) = {u ∈ L6(R3) | ‖u‖H˙1 := ‖∇u‖L2 <∞}.111.7. A Perturbation of the 3D Energy Critical NLSBy scaling invariance, Wµ := SµW = µ1/2W (µx), for µ > 0, also sat-isfy (1.14), as do their negatives and spatial translates ±Wµ(·+a) (a ∈ R3).These functions (and their multiples) are well-known to be the only functionsrealizing the best constant appearing in the Sobolev inequality [4, 75]∫R3|u|6 ≤ C3(∫R3|∇u|2)3, C3 =∫R3 W6(∫R3 |∇W |2)3 = 1(∫R3 W6)2 ,where the last equality used∫ |∇W |2 = ∫ W 6 (as follows from (1.12)). Aclosely related statement is that W , together with its scalings, negatives andspatial translates, are the only minimizers of the energy under the Pohozaevconstraint (1.12) with ε = ω = 0:min{E0(u) | 0 6= u ∈ H˙1(R3), K0(u) = 0} = E0(W ) = E0(±Wµ(·+ a)),K0(u) =∫R3{|∇u|2 − |u|6} .(1.15)It follows that for solutions of (1.13) lying energetically ‘below’ W , E0(u) <E0(W ), the sets where K0(u) > 0 and where K0(u) < 0 are invariantfor (1.13). The celebrated result [54] showed that radially symmetric so-lutions in the first set scatter to 0, while those in the second set becomesingular in finite time (in dimensions 3, 4, 5). In this way, W plays a centralrole in classifying solutions of (1.13), and it is natural to think of W (to-gether with its scalings and spatial translates) as the ground states of (1.13).The assumption in [54] that solutions be radially symmetric was removed in[57] for dimensions n ≥ 5 and then for n = 4 in [32]. Removing the radialsymmetry assumption appears still open for n = 3. A characterization ofthe dynamics for initial data at the threshold E0(u0) = E0(W ) appears in[33], and a classification of global dynamics based on initial data slightlyabove the ground state is given in [60].Just as the main interest in studying (1.13) is in exploring the implica-tions of critical scaling, the main interest in studying (1.10) and (1.11) hereis the effect of perturbing the critical scaling, in particular: the emergence ofground state solitary waves from the static solution W , the resulting energylandscape, and its implications for the dynamics.A natural analogue for (1.11) of the ground state variational problem (1.15)ismin{Sε,ω(u) | u ∈ H1 \ {0},Kε(u) = 0}. (1.16)For a study of similar minimization problems see [7] and [8] as well as [3],which treats a large class of critical problems and establishes the existence121.7. A Perturbation of the 3D Energy Critical NLSof ground state solutions. In space dimensions 4 and 5, [1, 2] showed theexistence of minimizers for (the analogue of) (1.16), hence of ground statesolitary waves, for each ω > 0 and εg(|u|2)u sufficiently small and subcritical;moreover, a blow-up/scattering dichotomy ‘below’ the ground states in thespirit of [54] holds. Our intention is to establish the existence of groundstates, and the blow-up/scattering dichotomy, in the 3-dimensional setting.In dimension 3, the question of the existence of minimizers for (1.16) is moresubtle, and we proceed via a perturbative construction, rather than a directvariational method.A key role in the analysis is played by the linearization of (1.14) aroundW , in particular the linearized operatorH := −∆ + V := −∆− 5W 4, (1.17)which as a consequence of scaling invariance has the following resonance:H ΛW = 0, ΛW :=ddµSµW |µ=0 =(12+ x · ∇)W /∈ L2(R3). (1.18)Indeed ΛW = W 3 − 12W decays like |x|−1, and soW, ΛW ∈ Lr(R3) ∩ H˙1(R3), 3 < r ≤ ∞.Our first goal is to find solutions to (1.11) where ω = ω(ε) > 0 is smalland Q(x) ∈ R is a perturbation of W in some appropriate sense. Oneobstacle is that W /∈ L2 is a slowly decaying function, whereas solutionsof (1.11) satisfy Q ∈ L2, and indeed are exponentially decaying.Assumption 1.7.1. Take f : R→ R ∈ C1 such that f(0) = 0 and|f ′(s)| . |s|p1−1 + |s|p2−1with 2 < p1 ≤ p2 <∞. Further assume that〈ΛW, f(W )〉 < 0.Theorem 1.7.2. There exists ε0 > 0 such that for each 0 < ε ≤ ε0, thereis ω = ω(ε) > 0, and smooth, real-valued, radially symmetric Q = Qε ∈H1(R3) satisfying (1.11) withω = ω1ε2 + ω˜ (1.19)Q(x) = W (x) + η(x) (1.20)131.7. A Perturbation of the 3D Energy Critical NLSwhereω1 =(−〈ΛW, f(W )〉6pi)2,ω˜ = O(ε2+δ1) for any δ1 < min(1, p1−2), ‖η‖Lr . ε1−3/r for all 3 < r ≤ ∞,and ‖η‖H˙1 . ε1/2. In particular, Q→W in Lr ∩ H˙1 as ε→ 0.Remark 1.7.3. We have a further decomposition of η but the leading orderterm depends on whether we measure it in Lr with r = ∞ or 3 < r < ∞.See Lemmas 3.1.9 and 3.1.10.Remark 1.7.4. Note that allowable f include f(Q) = |Q|p−1Q with 2 <p < 5, the subcritical, pure-power, focusing nonlinearities, as well as f(Q) =−|Q|p−1Q with 5 < p <∞, the supercritical, pure power, defocusing nonlin-earities. Observe〈ΛW,W p〉 =∫ (12W p+1 +W p(x · ∇)W)=∫ (12W p+1 +1p+ 1(x · ∇)W p+1)=∫ (12− 3p+ 1)W p+1which is negative when 2 < p < 5 and positive when p > 5.Remark 1.7.5. Since Qε → W in Lr for r ∈ (3,∞], the Pohozaev iden-tity (1.12), together with the divergence theorem, implies that for any suchfamily of solutions, a necessary condition is〈ΛW, f(W )〉 =∫ (12Wf(W )− 3F (W ))= limε→0∫ (12Qεf(Qε)− 3F (Qε))≤ 0.Remark 1.7.6. Note that Q ∈ Lr ∩ H˙1 (3 < r ≤ ∞) satisfying (1.11) liesautomatically in L2 (and hence H1): by the Pohozaev relations (1.12):0 =∫|∇Q|2 −∫Q6 − ε∫f(Q)Q+ ω∫Q2. (1.21)The first two integrals are then finite. We can also bound the third∣∣∣∣∫ f(Q)Q∣∣∣∣ ≤ ∫ |f(Q)||Q| . ∫ |Q|p1+1 + ∫ |Q|p2+1 <∞141.7. A Perturbation of the 3D Energy Critical NLSsince p2 + 1 ≥ p1 + 1 > 3. In this way∫Q2 must be finite. Moreover,since Q ∈ Lr with r > 6, a standard elliptic regularity argument impliesthat Q is in fact a smooth function. Therefore it suffices to find a solutionQ ∈ Lr ∩ H˙1.The paper [31] considers an elliptic problem similar to (1.11):−∆Q+Q−Qp − λQq = 0with 1 < q < 3, λ > 0 large and fixed, and p < 5 but p → 5. They demon-strate the existence of three positive solutions, one of which approaches W(1.14) as p → 5. The follow up [18] established a similar result with p → 5but p > 5 and 3 < q < 5. While [31] and [18] are perturbative in nature,their method of construction differs from ours.The proof of Theorem 1.7.2 is presented in Section 3.1. As the state-ment suggests, the argument is perturbative – the solitary wave profiles Qare constructed as small (in Lr) corrections to W . The set-up is given inSection 3.1.1. The equation for the correction η involves the resolvent ofthe linearized operator H. A Lyapunov-Schmidt-type procedure is used torecover uniform boundedness of this resolvent in the presence of the reso-nance ΛW – see Section 3.1.2 for the relevant estimates – and to determinethe frequency ω, see Section 3.1.3. Finally, the correction η is determinedby a fixed point argument in Section 3.1.4.The next question is if the solution Q is a ground state in a suitablesense. For this question, we will specialize to pure, subcritical powers f(Q) =|Q|p−1Q, 3 < p < 5, for which the ‘ground state’ variational problem (1.16)readsmin{Sε,ω(u) | u ∈ H1(R3) \ {0},Kε(u) = 0},Sε,ω(u) = 12‖∇u‖2L2 −16‖u‖6L6 −1(p+ 1)ε‖u‖p+1Lp+1+12ω‖u‖2L2 ,Kε(u) = ‖∇u‖2L2 − ‖u‖6L6 −3(p− 1)2(p+ 1)ε‖u‖p+1Lp+1.(1.22)Theorem 1.7.7. Let f(Q) = |Q|p−1Q with 3 < p < 5. There exists ε0 suchthat for each 0 < ε ≤ ε0 and ω = ω(ε) > 0 furnished by Theorem 1.7.2,the solitary wave profile Qε constructed in Theorem 1.7.2 is a minimizerof problem (1.22). Moreover, Qε is the unique positive, radially-symmetricminimizer.Remark 1.7.8. It follows from Theorem 1.7.7 that the solitary wave profilesare positive: Qε(x) > 0.151.7. A Perturbation of the 3D Energy Critical NLSRemark 1.7.9. (see Corollary 3.2.12). By scaling, for each ε > 0 there isan interval [ω,∞) 3 ω(ε), such that for ω ∈ [ω,∞),Q(x) :=(εεˆ) 15−pQεˆ((εεˆ) 25−px),where 0 < εˆ ≤ ε0 satisfies (ω(εˆ)/ω) = (εˆ/ε)4/(5−p), solves the correspondingminimization problem (1.22). Here the function Qεˆ is the solution con-structed by Theorem 1.7.2 with εˆ and ω(εˆ).The proof of Theorem 1.7.7 is presented in Section 3.2. It is somewhatindirect. We first use the Q = Qε constructed in Theorem 1.7.2 simply astest functions to verifySε,ω(ε)(Qε) < E0(W )and so confirm, by standard methods, that the variational problems (1.22)indeed admit minimizers. By exploiting the unperturbed variational prob-lem (1.15), we show these minimizers approach (up to rescaling) W as ε→ 0.Then the local uniqueness provided by the fixed-point argument from The-orem 1.7.2 implies that the minimizers agree with Qε.Finally, as in [1, 2], we use the variational problem (1.22) to character-ize the dynamics of radially-symmetric solutions of the perturbed criticalNonlinear Schro¨dinger equation{i∂tu = −∆u− |u|4u− ε|u|p−1uu(x, 0) = u0(x) ∈ H1(R3) (1.23)‘below the ground state’, in the spirit of [54]. By standard local existencetheory (details in Section 3.3), the Cauchy problem (2.1) admits a uniquesolution u ∈ C([0, Tmax);H1(R3)) on a maximal time interval, and a centralquestion is whether the solution blows-up in finite time (Tmax < ∞) or isglobal (Tmax =∞), and if global, how it behaves as t→∞. We have:Theorem 1.7.10. Let 3 < p < 5 and 0 < ε < ε0, let u0 ∈ H1(R3) beradially-symmetric, and satisfySε,ω(ε)(u0) < Sε,ω(ε)(Qε),and let u be the corresponding solution to (2.1):1. If Kε(u0) ≥ 0, u is global, and scatters to 0 as t→∞;2. if Kε(u0) < 0, u blows-up in finite time .161.7. A Perturbation of the 3D Energy Critical NLSNote that the conclusion is sharp in the sense that Qε itself is a globalbut non-scattering solution. Below the action of the ground state the setswhere Kε(u) > 0 and Kε(u) < 0 are invariant under the equation (1.10).Despite the fact that Kε(u0) > 0 gives an a priori bound on the H1 normof the solution, the local existence theory is insufficient (since we have theenergy critical power) to give global existence/scattering, and so we employconcentration compactness machinery.The blow-up argument is classical, while the proof of the scattering resultrests on that of [54] for the unperturbed problem, with adaptations to handlethe scaling-breaking perturbation coming from [1, 2] (higher-dimensionalcase) and [56] (defocusing case). This is given in Section 3.3.17Chapter 2A Degenerate EdgeBifurcation in the 1DLinearized NLSIn this chapter, we state and prove the theorem alluded to in Section 1.6.The problem is set up in Section 2.1. In Section 2.2 we collect some resultsabout the relevant operators that are necessary for the bifurcation analy-sis. Section 2.3 is devoted to the statement and proof of the main resultof this chapter: Theorem 2.3.1. The positivity of a certain (explicit) coef-ficient, which is crucial to the proof, is verified numerically; details of thiscomputation are given in Section Setup of the Birman-Schwinger ProblemWe consider the focusing, pure power (NLS) in one space dimension:i∂tu = −∂2xu− |u|p−1u. (2.1)Here u = u(x, t) : R × R → C with 1 < p < ∞. The NLS (2.1) admitssolutions of the formu(x, t) = Qp(x)eit (2.2)where Qp(x) > 0 satisfies−Q′′p −Qpp +Qp = 0. (2.3)In one dimension the explicit solutionsQp−1p (x) =(p+ 12)sech2(p− 12x)(2.4)of (2.3) for each p ∈ (1,∞) are classically known to be the unique H1solutions of (2.3) up to spatial translation and phase rotation (see e.g. [15]).182.1. Setup of the Birman-Schwinger ProblemIn what follows we study the linearized NLS problem. That is, linearize(2.1) about the solitary wave solutions (2.2) by considering solutions of theformu(x, t) = (Qp(x) + h(x, t)) eit.Then h solves, to leading order (i.e. neglecting terms nonlinear in h)i∂th = (−∂2x + 1)h−Qp−1p h− (p− 1)Qp−1p Re(h).We write the above as a matrix equation∂t~h = JHˆ~hwith~h :=(Re(h)Im(h))J−1 :=(0 −11 0)Hˆ :=(−∂2x + 1− pQp−1p 00 −∂2x + 1−Qp−1p).The above JHˆ is the linearized operator as it appears in [17]. We nowconsider the system rotatedi∂t~h = iJHˆ~hand find U unitary so that, UiJHˆU∗ = σ3H, where σ3 is one of the Paulimatrices and with H self-adjoint:σ3 =(1 00 −1), U =1√2(1 i1 −i),H =(−∂2x + 1 00 −∂2x + 1)− 12(p+ 1 p− 1p− 1 p+ 1)Qp−1p =: H˜ − V (p).In this way we are consistent with the formulation of [28, 84]. We can alsoarrive at this system, i∂t~h = σ3H~h, by letting ~h =(h h¯)Tfrom the start.Thus we are interested in the spectrum ofLp := σ3H192.1. Setup of the Birman-Schwinger Problemand so in what follows we consider the eigenvalue problemLpu = zu, z ∈ C, u ∈ L2(R,C2). (2.5)That the essential spectrum of Lp isσess(Lp) = (−∞,−1] ∪ [1,∞)and 0 is an eigenvalue of Lp are standard facts [17].When p = 3 we have the following resonance at the threshold z = 1 [17]u0 =(2−Q23−Q23)= 2(tanh2 x− sech2 x)(2.6)in the sense thatL3u0 = u0, u0 ∈ L∞, u0 /∈ Lq, for q <∞. (2.7)Our main interest is how this resonance bifurcates when p 6= 3 but |p− 3| issmall. We now seek an eigenvalue of (2.5) in the following formz = 1− α2, α > 0. (2.8)We note that the spectrum of Lp for the soliton (2.4) may only be located onthe Real or Imaginary axes [17], and so any eigenvalues in the neighbourhoodof z = 1 must be real. There is also a resonance at z = −1 which we do notmention further; symmetry of the spectrum of Lp ensures the two resonancesbifurcate in the same way.We now recast the problem in accordance with the Birman-Schwingerformulation (pp. 85 of [43]), as in [28, 84]. For (2.8), (2.5) becomes(σ3H˜ − 1 + α2)u = σ3V (p)u.The constant-coefficient operator on the left is now invertible so we can writeu = (σ3H˜ − 1 + α2)−1σ3V (p)u =: R(α)V (p)u.After noting that V (p) is positive we setw := V1/20 u, V0 := V(p=3)and apply V1/20 to arrive at the problemw = −Kα,pw, Kα,p := −V 1/20 R(α)V (p)V −1/20 (2.9)202.1. Setup of the Birman-Schwinger ProblemwithR(α) =((−∂2x + α2)−1 00 (−∂2x + 2− α2)−1). (2.10)We now seek solutions (α,w) of (2.9) which correspond to eigenvalues 1−α2and eigenfunctions V−1/20 w of (2.5). The decay of the potential V(p) andhence V120 now allows us to work in the space L2 = L2(R,C2), whose standardinner product we denote by 〈·, ·〉.The resolvent R(α) has integral kernelR(α)(x, y) =(12αe−α|x−y| 00 12√2−α2 e−√2−α2|x−y|)for α > 0. We expand R(α) asR(α) =1αR−1 +R0 + αR1 + α2RR. (2.11)These operators have the following integral kernelsR−1(x, y) =(12 00 0)R0(x, y) =(− |x−y|2 00 e−√2|x−y|2√2)R1(x, y) =(|x−y|24 00 0)and for α > 0 the remainder term RR is continuous in α and uniformlybounded as an operator from a weighted L2 space (with sufficiently strongpolynomial weight) to its dual. Moreover, since the entries of the full integralkernel R(α)(x, y) are bounded functions of |x− y|, we see that the entries ofRR(x, y) =1α2(R(α)(x, y)− ( 1αR−1(x, y) +R0(x, y) + αR1(x, y)))grow at most quadratically in |x − y| as |x − y| → ∞. We also expand thepotential V (p) in ε where ε := p− 3V (p) = V0 + εV1 + ε2V2 + ε3VR, ε := p− 3 (2.12)212.2. The Perturbed and Unperturbed OperatorsandV0 =(2 11 2)Q23 V1 =12(1 11 1)Q23 +(2 11 2)q1V2 =12(1 11 1)q1 +(2 11 2)q2 VR =12(1 11 1)q2 +(2 11 2)qRV1/20 =12(√3 + 1√3− 1√3− 1 √3 + 1)Q3.Here we have expandedQp−1p (x) = Q23(x) + εq1(x) + ε2q2(x) + ε3qR(x)and the computation givesQ23(x) = 2 sech2 x, q1(x) = sech2 x(12− 2x tanhx)q2(x) =12(2x2 tanh2 x sech2 x− x2 sech4 x− x tanhx sech2 x) .By Taylor’s theorem, the remainder term qR(x) satisfies an estimate of theform |qR(x)| ≤ C(1 + |x|3) sech2(x/2) for some constant C which is uniformin x and ε ∈ (−1, 1). We will henceforth writeQ for Q3 and Kα,ε for Kα,p.2.2 The Perturbed and Unperturbed OperatorsWe study (2.9), that is:(Kα,ε + 1)w = 0. (2.13)Using the expansions (2.11) and (2.12) for R(α) and V (p) we make the fol-lowing expansionKα,ε =1α(K−10 + εK−11 + ε2K−12 + ε3KR1)+K00 + εK01 + ε2K02 + ε3KR2+ αK10 + αεKR3+ α2KR4(2.14)where KR4 is uniformly bounded and continuous in α > 0 and ε in a neigh-bourhood of 0, as an operator on L2(R,C2).Before stating the main theorem we assemble some necessary facts aboutthe above operators.222.2. The Perturbed and Unperturbed OperatorsLemma 2.2.1. Each operator appearing in the expansion (2.14) for Kα,εis a Hilbert-Schmidt (so in particular bounded and compact) operator fromL2(R,C2) to itself.Proof. This is a straightforward consequence of the spatial decay of theweights which surround the resolvent. The facts that ‖V −1/20 ‖ ≤ Ce|x|, andthat ‖V 1/20 ‖ ≤ Ce−|x|, while each of ‖V0‖, ‖V1‖, ‖V2‖ and ‖VR‖ can bebounded by Ce−3|x|/2 (say if we restrict to |ε| < 12) imply easily that theseoperators all have square integrable integral kernels.Remark 2.2.2. The same decay estimates for the potentials used in theproof of Lemma 2.2.1 show that for α > 0 and w ∈ L2 solving (2.9) the corre-sponding eigenfunction of (2.5) u = V−1/20 w lies in L2 and so the eigenvaluez = 1−α2 is in fact a true eigenvalue. Indeed w ∈ L2 =⇒ V (p)V −1/20 w ∈ L2and so u = −R(α)V (p)V −1/20 w ∈ L2, since the free resolvent R(α) preservesL2 for α > 0 .We will also need the projections P and P which are defined as follows:for f ∈ L2 letPf :=〈v, f〉v‖v‖2 , v := V1/20(10)as well as the complementary P := 1−P . A direct computation shows thatfor any f ∈ L2 we haveK−10f = −4Pf. (2.15)Note that all operators in the expansion containing R−1 return outputs inthe direction of v.Lemma 2.2.3. The operator P (K00 + 1)P has a one dimensional kernelspanned byw0 := V1/20 u0as an operator from Ran(P ) to Ran(P ).Proof. First note that by (2.7)−V0u0 = σ3u0 − H˜u0, [−V0u0]1 = [u0]′′1 (2.16)232.2. The Perturbed and Unperturbed Operatorsfrom which it follows thatPw0 = 0, i.e. w0 ∈ Ran(P ).Then a direct computation using (2.16), the expansion (2.14), the expressionfor R0, and integration by parts, shows that(K00 + 1)w0 = 2vand so indeed P (K00 + 1)Pw0 = 0.Theorem 5.2 in [48] shows that the kernel of the analogous scalar operatorcan be at most one dimensional. We will use this argument, adapted to thevector structure, to show that any two non-zero elements of the kernel mustbe multiples of each other. Take w ∈ L2 with 〈w, v〉 = 0 and P (K00 +1)w =0. That is (K00 + 1)w = cv for some constant c. This means−V 1/20 R0V0V −1/20 w + w = cV 1/20(10).Let w = V1/20 u where u =(u1u2). We then obtain, after rearranging andexpanding(u1u2)=(c− 12∫R |x− y|Q2(y) (2u1(y) + u2(y)) dy12√2∫R exp(−√2|x− y|)Q2(y)(u1(y) + 2u2(y))dy).We now rearrange the first component. Expand−12∫R|x− y|Q2(y)(2u1(y) + u2(y))dy=− 12∫ x−∞(x− y)Q2(y)(2u1(y) + u2(y))dy− 12∫ ∞x(y − x)Q2(y)(2u1(y) + u2(y))dyand rewrite the first term as− x2∫ x−∞Q2(y)(2u1(y) + u2(y))dy +12∫ x−∞yQ2(y)(2u1(y) + u2(y))dy=x2∫ ∞xQ2(y)(2u1(y) + u2(y))dy + b− 12∫ ∞xyQ2(y)(2u1(y) + u2(y))dy242.2. The Perturbed and Unperturbed Operatorswhereb :=12∫RyQ2(y)(2u1(y) + u2(y))dyand where we used∫R 2Q2u1 + Q2u2 = 0 since 〈w, v〉 = 0. So puttingeverything back together we see(u1u2)=(c+ b+∫∞x (x− y)Q2(y) (2u1(y) + u2(y)) dy12√2∫R exp(−√2|x− y|)Q2(y)(u1(y) + 2u2(y))dy). (2.17)We claim that as x→∞(u1u2)→(c+ b0).Observe ∣∣∣∣ ∫ ∞x(x− y)Q2(y) (2u1(y) + u2(y)) dy∣∣∣∣≤∫ ∞x|y − x|Q2(y)|2u1(y) + u2(y)|dy≤∫ ∞x|y|Q2(y)|2u1(y) + u2(y)|dy→ 0as x→∞. Here we have used the fact that w ∈ L2 implies Q|2u1 +u2| ∈ L2and that |y|Q ∈ L2. As well, in the second component∫Re−√2|x−y|Q2(y)(u1(y) + 2u2(y))dy=e−√2x∫ x−∞e√2yQ2(y)(u1(y) + 2u2(y))dy+ e√2x∫ ∞xe−√2yQ2(y)(u1(y) + 2u2(y))dy252.2. The Perturbed and Unperturbed Operatorsand∣∣∣∣e−√2x ∫ x−∞ e√2yQ2(y)(u1(y) + 2u2(y))dy∣∣∣∣≤ e−√2x∫ x−∞e√2yQ2(y)|u1(y) + 2u2(y)|dy≤ e−√2x(∫ x−∞e2√2yQ2(y)dy)1/2(∫ x−∞Q2(y)|u1(y) + 2u2(y)|2dy)1/2≤ Ce−√2x(∫ x−∞e2√2yQ2(y)dy)1/2≤ Ce−√2x(∫ x−∞e2√2ye−2ydy)1/2≤ Ce−√2x(e−2√2xe−2x)1/2 ≤ Ce−x → 0, x→∞where we again used Q|u1 + 2u2| ∈ L2. Similarly,∣∣∣∣e√2x ∫ ∞xe−√2yQ2(y)(u1(y) + 2u2(y))dy∣∣∣∣→ 0as x→∞ which addresses the claim.Next we claim that if c + b = 0 in (2.17) then u ≡ 0. To address theclaim we first note that if c + b = 0 then u ≡ 0 for all x ≥ X for some X,by estimates similar to those just done. Finally, we appeal to ODE theory.Differentiating (2.17) in x twice returns the systemu′′1 = −2Q2u1 −Q2u2 (2.18)u′′2 − 2u2 = −Q2u1 − 2Q2u2. (2.19)Any solution u to the above with u ≡ 0 for all large enough x must beidentically zero.With the claim in hand we finish the argument. Given two non-zeroelements of the kernel, say u and u˜ with limits as x→∞ (written as above)c+ b and c˜+ b˜ respectively, the combinationu∗ = u− c+ bc˜+ b˜u˜satisfies (2.17) but with u∗(x) → 0 as x → ∞, and so u∗ ≡ 0. Therefore, uand u˜ are linearly dependent, as required.262.3. Bifurcation AnalysisNote that K00, and hence P (K00 + 1)P , is self-adjoint. IndeedK00 = −V 1/20 R0V0V −1/20= −V −1/20 V0R0V 1/20= (K00)∗.As we have seen above in Lemma 2.2.1, thanks to the decay of the poten-tial, PK00P is a compact operator. Therefore, the simple eigenvalue −1 ofPK00P is isolated and so(P (K00 + 1)P )−1 : {v, w0}⊥ → {v, w0}⊥ (2.20)exists and is bounded.With the above preliminary facts assembled, we proceed to the bifurca-tion analysis.2.3 Bifurcation AnalysisThis section is devoted to the proof of the main result of Chapter 2:Theorem 2.3.1. There exists ε0 > 0 such that for −ε0 ≤ ε ≤ ε0 theeigenvalue problem (2.13) has a solution (α,w) of the formw = w0 + εw1 + ε2w2 + w˜α = ε2α2 + α˜(2.21)where α2 > 0, w0, w1, w2 are known (given below), and |α˜| < C|ε|3 and‖w˜‖L2 < C|ε|3 for some C > 0.Remark 2.3.2. This theorem confirms the behaviour observed numericallyin [17]: for p 6= 3 but close to 3, the linearized operator JHˆ (which is uni-tarily equivalent to iLp) has true, purely imaginary eigenvalues in the gapbetween the branches of essential spectrum, which approach the thresholdsas p → 3. Note Remark 2.2.2 to see that u = V −1/20 w is a true L2 eigen-function of (2.5). In addition, the eigenfunction approaches the resonanceeigenfunction in some weighted L2 space. Furthermore, we have found thatα2, the distance of the eigenvalues from the thresholds, is to leading orderproportional to (p − 3)4. Finally, note that α = ε2α2 + O(ε3) with α2 > 0gives α > 0 for both ε > 0 and ε < 0, ensuring the eigenvalues appear onboth sides of p = 3.272.3. Bifurcation AnalysisThe quantities in (2.21) are defined as follows:w0 := V1/20 u0Pw1 :=14K−11w0Pw1 := −(P (K00 + 1)P)−1(14PK00K−11w0 + PK01w0)Pw2 :=14(K−11w1 +K−12w0 + α2(K00 + 1)w0)Pw2 := −(P (K00 + 1)P)−1(14PK00K−11w1 +14PK00K−12w0+α24PK00(K00 + 1)w0 + PK01w1 + PK02w0 + α2PK10w0)α2 :=−14〈w0,K00K−11w1〉 − 14〈w0,K00K−12w0〉 − 〈w0,K01w1〉 − 〈w0,K02w0〉〈w0,K10w0〉+ 14〈w0,K00(K00 + 1)w0〉.Remark 2.3.3. A numerical computation showsα2 ≈ 2.52/8 > 0.Since the positivity of α2 is crucial to the main result, details of this com-putation are described in Section 2.4.Note that the functions on which P (K00 + 1)P is being inverted in theexpressions for Pw1 and Pw2 are orthogonal to both w0 and v, and so thesequantities are well-defined by (2.20). The projections to v are zero by thepresence of P . As for the projections to w0, the identity〈w0, 14K00K−11w0 +K01w0〉 = 0 (2.22)has been verified analytically. It is because of this identity that the O(ε)term is absent in the expansion of α in (2.21). The fact that0 = 〈w0, 14K00K−11w1 +14K00K−12w0 +α24K00(K00 + 1)w0 +K01w1+K02w0 + α2K10w0〉comes from our definition of α2.282.3. Bifurcation AnalysisThe above definitions, along with (2.15), imply the relationships below0 = K−10w0 (2.23)0 = K−11w0 +K−10w1 (2.24)0 = K−10w2 +K−11w1 +K−12w0 + α2(K00 + 1)w0 (2.25)0 = P (K00 + 1)w1 + PK01w0 (2.26)0 = P (K00 + 1)w2 + PK01w1 + PK02w0 + α2PK10w0 (2.27)which we will use in what follows.Using the expression for α in (3.1.1), our expansion (2.14) for Kα,ε nowtakes the formKα,ε =1α(K−10 + εK−11 + ε2K−12 + ε3KR1)+K00 + εK01 + ε2K02 + ε3KR2+ (α2ε2 + α˜)K10 + (α2ε2 + α˜)εKR3 + (α2ε2 + α˜)2KR4=:1α(K−10 + εK−11 + ε2K−12 + ε3KR1)+K00 + εK1 + α˜K2where K1 is a bounded (uniformly in ε) operator depending on ε but not α˜,while K2 is a bounded (uniformly in ε and α˜) operator depending on bothε and α˜.Further decomposingw˜ = βv +W, 〈W, v〉 = 0,we aim to show existence of a solution with the remainder terms α˜, β andW small. We do so via a Lyapunov-Schmidt reduction.First substitute (2.21) to (2.13) and apply the projection P to obtain0 = P (Kα,ε + 1)w= P (Kα,ε + 1)(w0 + εw1 + ε2w2 + βv +W )= P (K00 + 1)w0 + εP (K00 + 1)w1 + εPK01w0+ ε2P (K00 + 1)w2 + ε2PK01w1 + ε2PK02w0 + ε2α2PK10w0+ P (K00 + 1)(βv +W ) + α˜PK10w0 + P(εK1 + α˜K2)(βv +W )+ ε3P(KR2w0 +K02w1 +K01w2 + εK02w2 + εKR2w1 + ε2KR2w2)+ (α2ε2 + α˜)PK10(εw1 + ε2w2) + (α2ε2 + α˜)εPKR3(w0 + εw1 + ε2w2)+ (α2ε2 + α˜)2PKR4(w0 + εw1 + ε2w2).(2.28)292.3. Bifurcation AnalysisMaking some cancellations coming from Lemma 2.2.3, (2.26) and (2.27)leads to−P (K00 + 1)PW =βPK00v + α˜PK10w0 + P(εK1 + α˜K2)(βv +W )+ ε3P(KR2w0 +K02w1 +K01w2 + εK02w2 + εKR2w1 + ε2KR2w2)+ (α2ε2 + α˜)PK10(εw1 + ε2w2) + (α2ε2 + α˜)εPKR3(w0 + εw1 + ε2w2)+ (α2ε2 + α˜)2PKR4(w0 + εw1 + ε2w2)=: F(W ; ε, α˜, β).According to (2.20), inversion of P (K00 + 1)P on F requires the solv-ability conditionP0F = 0, P0 := 1‖w0‖22〈w0, ·〉w0, P 0 := 1− P0 (2.29)which we solve together with the fixed point problemW =(−P (K00 + 1)P )−1 P 0F(W ; ε, α˜, β) =: G(W ; ε, α˜, β) (2.30)in order to solve (2.28).WriteF := P (βK00v + α˜K10w0 + (εK1 + α˜K2) (βv +W ) + ε3f1 + εα˜f2 + α˜2h1)where f1 and f2 denote functions depending on (and L2 bounded uniformlyin) ε but not α˜, while h1 denotes an L2 function depending on (and uniformlyL2 bounded in) both ε and α˜.Lemma 2.3.4. For any M > 0 there exists ε0 > 0 and R > 0 such that forall −ε0 ≤ ε ≤ ε0 and for all α˜ and β with |α˜| ≤M |ε|3 and |β| ≤M |ε|3 thereexists a unique solution W ∈ L2 ∩ {v, w0}⊥ of (2.30) satisfying ‖W‖L2 ≤R|ε|3.Proof. We prove this by means of Banach Fixed Point Theorem. We mustshow that G(W ) maps the closed ball of radius R|ε|3 into itself and thatG(W ) is a contraction mapping. Taking W ∈ L2 orthogonal to v and w0 suchthat ‖W‖L2 ≤ R|ε|3 and given M > 0 where |α˜| ≤ M |ε|3 and |β| ≤ M |ε|3,302.3. Bifurcation Analysiswe have, using the boundedness of(−P (K00 + 1)P )−1 P 0,‖G‖L2≤ C|β|‖PK00v + P(εK1 + α˜K2)v‖L2 + C|α˜|‖P (K10w0 + εf2 + α˜h1) ‖L2+ C‖P (εK1 + α˜K2)W‖L2 + |ε|3C‖Pf1‖L2≤ CM |ε|3 + CM |ε|3 + C|ε|‖W‖L2 + C|α˜|‖W‖L2 + C|ε|3≤ C|ε|3 + CR|ε|4≤ R|ε|3for some appropriately chosen R with |ε| small enough. Here C is a positive,finite constant whose value changes at each appearance. Next consider‖G(W1)− G(W2)‖L2≤ C‖P (εK1 + α˜K2) ‖L2→L2‖W1 −W2‖L2≤ C|ε|‖P K1‖L2→L2‖W1 −W2‖L2 + C|α˜|‖P K2‖L2→L2‖W1 −W2‖L2≤ C|ε|‖W1 −W2‖L2 ≤ κ‖W1 −W2‖L2with 0 < κ < 1 by taking |ε| sufficiently small. Hence G(W ) is a contraction,and we obtain the desired result.Lemma 2.3.4 provides W as a function of α˜ and β, which we may thensubstitute into (2.29) to get0 = 〈w0,F〉= β〈w0,K00v〉+ α˜〈w0,K10w0〉+ εβ〈w0,K1v〉+ α˜β〈w0,K2v〉+ ε3〈w0, f1〉+ εα˜〈w0, f2〉+ α˜2〈w0, h1〉+ ε〈w0,K1W 〉+ α˜〈w0,K2W 〉=: β〈w0,K00v〉+ α˜〈w0,K10w0〉+ F1 (2.31)which is the first of two equations relating α˜ and β.The second equation is the complementary one to (2.28): substitute312.3. Bifurcation Analysis(2.21) to (2.13) but this time multiply by α and take projection P to see0 = αP (Kα,ε + 1)w= K−10w0 + ε(K−11w0 +K−10w1)+ ε2 (K−10w2 +K−11w1 +K−12w0) + ε2α2(K00 + 1)w0+ ε3(K−11w2 +K−12w1 +KR1w0 + εK−12w2 + εKR1w1 + ε2KR1w2)+ βK−10v +K−10W + ε(K−11 + εK−12 + ε2KR1)(βv +W )+ α˜(K00 + 1)w0 + ε3α2P (K00 + 1)(w1 + εw2) + εα˜P (K00 + 1)(w1 + εw2)+ ε2α2P (K00 + 1)(βv +W ) + α˜P (K00 + 1)(βv +W )+ αP (εK01 + ε2K02 + ε3KR2 + αK10 + αεKR3 + α2KR4)× (w0 + εw1 + ε2w2 + βv +W ).(2.32)After using known information about w0, w1, w2, α2 coming from (2.23),(2.24), (2.25) and noting that K−10W = −4PW = 0 from (2.15) we have0 = βK−10v + α˜(K00 + 1)w0+ ε3(K−11w2 +K−12w1 +KR1w0 + εK−12w2 + εKR1w1 + ε2KR1w2)+ ε(K−11 + εK−12 + ε2KR1)(βv +W )+ ε3α2P (K00 + 1)(w1 + εw2) + εα˜P (K00 + 1)(w1 + εw2)+ ε2α2P (K00 + 1)(βv +W ) + α˜P (K00 + 1)(βv +W )+ αP (εK01 + ε2K02 + ε3KR2 + αK10 + αεKR3 + α2KR4)× (w0 + εw1 + ε2w2 + βv +W ).Written more compactly, this is0 =βK−10v + α˜(K00 + 1)w0+ ε3f4 + εK3(βv +W ) + α˜εf5 + α˜K4(βv +W ) + α˜2h2where K3 is a bounded (uniformly in ε) operator containing ε but not α˜,while K4 is a bounded (uniformly in ε and α˜) operator containing both εand α˜. Functions f4 and f5 depend on ε (and are uniformly L2-bounded)but not α˜, while the function h2 depends on both ε and α˜ (and is uniformlyL2-bounded). To make the relationship between α˜ and β more explicit we322.3. Bifurcation Analysistake inner product with v0 = β〈v,K−10v〉+ α˜〈v, (K00 + 1)w0〉+ ε3〈v, f4〉+ ε〈v,K3(βv +W )〉+ α˜ε〈v, f5〉+ α˜〈v,K4(βv +W )〉+ α˜2〈v, h2〉=: β〈v,K−10v〉+ α˜〈v, (K00 + 1)w0〉+ F2. (2.33)Now let~ζ =(α˜β)and rewrite (2.31) and (2.33) in the following wayA~ζ :=( 〈w0,K10w0〉 〈w0,K00v〉〈v, (K00 + 1)w0〉 〈v,K−10v〉)(α˜β)=(F1F2)which we recast as a fixed point problem~ζ = A−1(F1F2)=: ~F (α˜, β; ε). (2.34)We have computedA =(0 1616 −32)so in particular, A is invertible. We wish to show there is a solution (α˜, β)of (2.34) of the appropriate size. We establish this fact in the followingLemmas. Lemmas 2.3.5 and 2.3.6 are accessory to Lemma 2.3.7.Lemma 2.3.5. The operators and functions K2, K4 and h1, h2 are contin-uous in α˜ > 0.Proof. The operators and function in question are compositions of continu-ous functions of α˜.Lemma 2.3.6. The W given by Lemma 2.3.4 is continuous in ~ζ for suffi-ciently small |ε|.Proof. Let (α˜1, β1) give rise to W1 and let (α˜2, β2) give rise to W2 via Lemma2.3.4. Take |α˜1−α˜2| < δ and |β1−β2| < δ. We show that ‖W1−W2‖L2 < Cδ332.3. Bifurcation Analysisfor some constant C > 0. Observing K2 depends on α˜, we see‖W1 −W2‖L2 =‖ (P (K00 + 1)P )−1 P 0‖L2→L2‖F(W1, ~ζ1; ε)−F(W2, ~ζ2; ε)‖L2≤ C∥∥∥∥(β1 − β2)K00v + (α˜1 − α˜2)K10w0 + ε(β1 − β2)K1v+ εK1(W1 −W2) + α˜1β1K2(α˜1)v − α˜2β2K2(α˜2)v + α˜1K2(α˜1)W1− α˜2K2(α˜2)W2 + ε(α˜1 − α˜2)f2 + α˜21h1(α˜1)− α˜22h1(α˜2)∥∥∥∥L2≤ Cδ + C|ε|‖W1 −W2‖L2+ ‖α˜1K2(α˜1)(W1 −W2) +(α˜1K2(α˜1)− α˜2K2(α˜2))W2‖L2≤ Cδ + C|ε|‖W1 −W2‖L2noting that |α˜1| ≤M |ε|3. Rearranging the above gives‖W1 −W2‖L2 < Cδfor small enough |ε|.Lemma 2.3.7. There exists ε0 > 0 such that for all −ε0 ≤ ε ≤ ε0 theequation (2.34) has a fixed point with |α˜|, |β| ≤M |ε|3 for some M > 0.Proof. We prove this by means of the Brouwer Fixed Point Theorem. Weshow that ~F maps a closed square into itself and that ~F is a continu-ous function. Take |α˜|, |β| ≤ M |ε|3 and and so by Lemma 2.3.4 we have‖W‖L2 ≤ |ε|3R for some R > 0. Consider now‖A−1‖ |F1|≤ ‖A−1‖(|ε||β||〈w0,K1v〉|+ |α˜||β||〈w0,K2v〉|+ |ε|3|〈w0, f1〉|+ |ε||α˜||〈w0, f2〉|+ |α˜|2|〈w0, h1〉|+ |ε||〈w0,K1W 〉|+ |α˜||〈w0,K2W 〉|)≤ CM |ε|4 + CM2|ε|6 + C|ε|3 + CM |ε|4 + CM2|ε|6 + CR|ε|4≤ C|ε|3 + CM |ε|4 ≤M |ε|3342.3. Bifurcation Analysisand‖A−1‖ |F2|≤ ‖A−1‖(|ε|3|〈v, f4〉|+ |ε||〈v,K3(βv +W )〉|+ |α˜||ε||〈v, f5〉|+ |α˜||〈v,K4(βv +W )〉|+ |α˜|2|〈v, h2〉|)≤ C|ε|3 + CM |ε|4 + CR|ε|4 + CM |ε|4 + CM2|ε|6 + CMR|ε|6 + CM2|ε|6≤ C|ε|3 + CM |ε|4 ≤M |ε|3for some choice of M > 0 and sufficiently small |ε| > 0. Here C > 0 is aconstant that is different at each instant. So ~F maps the closed square toitself.It is left to show that ~F is continuous. Given η > 0 take |α˜1 − α˜2| < δand |β1 − β2| < δ. Let (α˜1, β1) give rise to W1 and let (α˜2, β2) give rise toW2 via Lemma 2.3.4. We will also use Lemma 2.3.5 and Lemma 2.3.6. Nowconsider|F1(α˜1, β1)−F1(α˜2, β2)|=∣∣∣ε(β1 − β2)〈w0,K1v〉+ α˜1β1〈w0,K2(α˜1)v〉 − α˜2β2〈w0,K2(α˜2)v〉+ ε(α˜1 − α˜2)〈w0, f2〉+ α˜21〈w0, h1(α˜1)〉 − α˜22〈w0, h1(α˜2)〉+ ε〈w0,K1(W1 −W2)〉+ α˜1〈w0,K2(α˜1)W1〉 − α˜2〈w0,K2(α˜2)W2〉∣∣∣≤ Cδ + C‖h1(α˜1)− h1(α˜2)‖L2+ C‖W1 −W2‖L2 + C‖K2(α˜1)−K2(α˜2)‖L2→L2≤ Cδ < η‖A−1‖√2for small enough δ. Similarly we can show|F2(α˜1, β1)−F2(α˜2, β2)| ≤ Cδ < η‖A−1‖√2for δ small enough. Putting everything together gives |~F (~ζ1) − ~F (~ζ2)| < ηas required. Hence ~F is continuous.So finally we have solved both (2.28) and (2.32), and hence (2.13), andso have proved Theorem Comments on the Computations2.4 Comments on the ComputationsAnalytical and numerical computations were used in the above to computeinner products such as the ones appearing in the definition of α2 (2.21). Itwas critical to establish that α2 > 0 since the expansion of the resolventR(α) (2.10) requires α > 0. Inner products containing w0 but not w1 canbe written as an explicit single integral and then evaluated analytically ornumerically with good accuracy. For example〈w0,K02w0〉+ 14〈w0,K00K−12w0〉=− 12∫R2|x− y|(4Q2(x)− 3Q4(x))× (Q2(y)q1(y)− q1(y) + 3Q2(y)q2(y)− 4q2(y)− c22Q2(y))dydx+12√2∫R2e−√2|x−y|(2Q2(x)− 3Q4(x))× (Q2(y)q1(y)− q1(y) + 3Q2(y)q2(y)− 2q2(y)− c24Q2(y))dydx=−∫RQ2(y)(Q2(y)q1(y)− q1(y) + 3Q2(y)q2(y)− 4q2(y)− c22Q2(y))dy−∫RQ2(y)(Q2(y)q1(y)− q1(y) + 3Q2(y)q2(y)− 2q2(y)− c24Q2(y))dy≈− 2.9369wherec2 =12∫RQ2q1 − q1 + 3Q2q2 − 4q2.To reduce the double integral to a single integral we recall some facts aboutthe integral kernels. Leth(y) = −12∫R|x− y|(4Q2(x)− 3Q4(x))dx.Then h solves the equationh′′ = −4Q2 + 3Q4.Notice that −4Q2 + 3Q4 = −2Q2u1−Q2u2 where u1 and u2 are the compo-nents of the resonance u0 (2.6). Observing the equation (2.18) we see thath = u1 + c = 2 − Q2 + c for some constant c. We can directly compute362.4. Comments on the Computationsh(0) = −2 to find c = −2 and so h = −Q2. A similar argument involving(2.19) gives12√2∫Re−√2|x−y|(2Q2(x)− 3Q4(x))dx = u2(y) = −Q2(y).Many of the inner products can be computed analytically. These includethe identity (2.22), the entires in the matrix A in (2.34) and the denomi-nator appearing in the expression for α2. As an example we evaluate thedenominator of α2:〈w0,K10w0〉+ 14〈w0,K00(K00 + 1)w0〉= −∫R2(3Q4(x)− 4Q2(x))(x− y)24(3Q4(y)− 4Q2(y))dydx+12∫R2(4Q2(x)− 3Q4(x))|x− y|Q2(y)dydx− 14√2∫R2(4Q2(x)− 3Q4(x))e−√2|x−y|Q2(y)dydx=32∫RQ4(y)dy= 8where the first integral is zero by a direct computation and the remainingdouble integrals are converted to single integrals as above.Computing inner products containing w1 is harder. We have an explicitexpression for Pw1 but lack an explicit expression for Pw1. Therefore weapproximate Pw1 by numerically inverting P (K00 + 1)P inP (K00 + 1)Pw1 = −(14PK00K−11w0 + PK01w0)=: g.Note that 〈g, v〉 = 〈g, w0〉 = 0. We represent P (K00 + 1)P as a matrix withrespect to a basis {φj}Nj=1. The basis is formed by taking terms from thetypical Fourier basis and projecting out the components of each functionin the direction of v and w0. Some basis functions were removed to ensurelinear independence of the basis. Let Pw1 =∑Nj=1 ajφj . ThenB~a = ~bwhere Bj,k = 〈φj , (K00 + 1)φk〉 and bj = 〈φj , g〉. So we can solve for ~a byinverting the matrix B. Once we have an approximation for Pw1 we can372.4. Comments on the Computations−6 −4 −2 0 2 4 6−1−0.8−0.6−0.4− o f Pw1  Figure 2.1: The two components of Pw1 computed numerically with 32basis terms.compute P (K00 + 1)Pw1 directly to observe agreement with the function g.With this agreement we are confident in our numerical algorithm and thatour numerical approximation for Pw1 is accurate. In Figure 2.1 we showthe two components of Pw1 as computed numerically. Figure 2.2 shows thecomponents of the function g with the computed P (K00 + 1)Pw1 on top.With an approximation for Pw1 in hand we can combine it with ourexplicit expression for Pw1 and compute inner products containing w1 inthe same way as the previous inner product containing w0. In this way weestablish that α2 > 0. We list computed values for the numerator of α2against the number of basis terms used in Table 2.1.382.4. Comments on the Computations−6 −4 −2 0 2 4 6−0.5−0.4−0.3−0.2− Consistency CheckFigure 2.2: The two components of function g with the computed P (K00 +1)Pw1 on top. Again 32 basis terms were used in this computation. At thisscale the difference can only be seen around zero and at the endpoints.Number of Basis Terms 8α220 2.499224 2.513728 2.518930 2.520132 2.5207Table 2.1: Numerical values for 8α2 for the number of basis terms used inthe computation.39Chapter 3Perturbations of the 3DEnergy Critical NLSIn this chapter we prove Theorems 1.7.2, 1.7.7, and 1.7.10. The constructionof the solitary wave profiles appears in Section 3.1, variational argumentswhich establish the solitary waves as ground states appear in Section 3.2,and the dynamical (scattering/blow-up) theory appears in Section Construction of Solitary Wave ProfilesThis section is devoted to the proof of Theorem 1.7.2, constructing solitarywave profiles for the perturbed NLS via perturbation from the unperturbedstatic solution W .3.1.1 Mathematical SetupLet λ2 = ω with λ ≥ 0. Now substitute (1.20) to (1.11) to see(−∆− 5W 4 + λ2)η = −λ2W + εf(W ) +N(η)whereN(η) = (W + η)5 −W 5 − 5W 4η + ε (f(W + η)− f(W ))collects the higher order terms. We can rewrite the above as(H + λ2)η = F , H = −∆ + V, V = −5W 4 (3.1)whereF = F(ε, λ, η) = −λ2W + εf(W ) +N(η).To understand the resolvent (H + λ2)−1 for small λ, we follow [47]. Use theresolvent identity to write(H + λ2)−1 = (1 +R0(−λ2)V )−1R0(−λ2)403.1. Construction of Solitary Wave ProfileswhereR0(ζ) = (−∆− ζ)−1is the free resolvent, and apply Lemma 4.3 of [47] to obtain the expansion(1 +R0(−λ2)V )−1 = − 1λ〈V ψ, ·〉ψ +O(1) (3.2)where ψ is the normalized resonance eigenfunction (1.18):ψ(x) =1√3piΛW (x),∫R3V ψ =√4pi.The above expansion is understood in [47] in weighted Sobolev spaces. Wechoose instead to work in higher Lp spaces. Precise statements are found inthe following Section 3.1.2.To eliminate the singular behaviour as λ→ 0 we require0 = 〈R0(−λ2)V ψ,F(ε, λ, η)〉. (3.3)Satisfying this condition determines λ = λ(ε, η). This is done in Section3.1.3. With this condition met, we can invert (3.1) to seeη = (H + λ2)−1F = (H + λ2(ε, η))−1F(ε, λ(ε, η), η) =: G(η, ε), (3.4)which can be solved for η via a fixed point argument. This is done inSection Resolvent EstimatesWe collect here some estimates that are necessary for the proof of Theorem1.7.2.In order to apply Lemma 4.3 of [47] and so to use the expansion (3.2)in what follows (Lemmas 3.1.1 and 3.1.4) we must have that the operatorH has no zero eigenvalue. However, it is true that H(∂W/∂xj) = 0 foreach j = 1, 2, 3. To this end, we restrict ourselves to considering only radialfunctions. In this way H has no zero eigenvalues and only the one resonance,ΛW (see [33]).The free resolvent operator R0(−λ2) for λ > 0 has integral kernelR0(−λ2)(x) = e−λ|x|4pi|x| . (3.5)413.1. Construction of Solitary Wave ProfilesAn application of Young’s inequality/generalized Young’s inequality givesthe bounds‖R0(−λ2)‖Lq→Lr . λ3(1/q−1/r)−2, 1 ≤ q ≤ r ≤ ∞ (3.6)‖R0(−λ2)‖Lqw→Lr . λ3(1/q−1/r)−2, 1 < q ≤ r <∞ (3.7)with 3(1/q − 1/r) < 2, as well as‖R0(−λ2)‖Lq→Lr . 1 (3.8)where 1 < q < 3/2 and 3(1/q − 1/r) = 2 (so 3 < r <∞). We will also needthe additional bound‖R0(−λ2)‖L32−∩L 32+→L∞ . 1, (3.9)where the +/− means the bound holds for any exponent greater/less than3/2, to replace the fact that we do not have (3.8) for r =∞ and q = 3/2.Observe also that R0(0) = G0 has integral kernelG0(x) =14pi|x|and is formally (−∆)−1.We need also some facts about the operator (1 + R0(−λ2)V )−1. Theidea is that we can think of the full resolvent (1 + R0(−λ2)V )−1R0(−λ2)as behaving like the free resolvent R0(−λ2) providing we have a suitableorthogonality condition. Otherwise we lose a power of λ due to the non-invertibility of (1 +G0V ): indeed,ψ ∈ ker(1 +G0V ), V ψ ∈ ker ((1 +G0V )∗ = 1 + V G0) . (3.10)First we recall some results of [47]:Lemma 3.1.1. (Lemmas 2.2 and 4.3 from [47]) Let s satisfy 3/2 < s < 5/2and denote B = B(H1−s, H1−s) where H1−s is the weighted Sobolev space withnorm‖u‖H1−s = ‖(1 + |x|2)−s/2u‖H1 .Then for ζ with Imζ ≥ 0 we have the expansions1 +R0(ζ)V = 1 +G0V + iζ1/2G1V + o(ζ1/2)(1 +R0(ζ)V )−1 = −iζ−1/2〈·, V ψ〉ψ + C10 + o(1)423.1. Construction of Solitary Wave Profilesin B with |ζ| → 0. Here C10 is an explicit operator and G0 and G1 areconvolution with the kernelsG0(x) =14pi|x| , G1(x) =14pi.Remark 3.1.2. The expansion is also valid in B(L2−s, L2−s) where L2−s isthe weighted L2 space with norm‖u‖L2−s = ‖(1 + |x|2)−s/2u‖L2 .Remark 3.1.3. Since our potential only has decay |V (x)| . 〈x〉−4 ourexpansion has one less term than in [47] and we use 3/2 < s < 5/2 ratherthan 5/2 < s < 7/2.The following is a reformulation of Lemma 3.1.1 but using higher Lpspaces rather than weighted spaces. This reformulation was also used in[44].Lemma 3.1.4. Take 3 < r ≤ ∞ and λ > 0 small. Then‖(1 +R0(−λ2)V )−1f‖Lr . 1λ‖f‖Lr .If we also have 〈V ψ, f〉 = 0 then‖(1 +R0(−λ2)V )−1f‖Lr . ‖f‖Lrand‖(1 +R0(−λ2)V )−1f − Q¯(1 +G0V )−1P¯ f‖Lr.{λ1−3/r, 3 < r <∞λ log(1/λ), r =∞}‖f‖Lr (3.11)whereP :=1∫V ψ2〈V ψ, ·〉ψ, P¯ = 1− PQ :=1∫V ψ〈V, ·〉ψ, Q¯ = 1−Q(3.12)Proof. We start with the identityg := (1 +R0(−λ2)V )−1f = f −R0(−λ2)V (1 +R0(−λ2)V )−1f= f −R0(−λ2)V g433.1. Construction of Solitary Wave Profilesso‖g‖Lr . ‖f‖Lr + ‖R0(−λ2)V g‖Lr .We treat the above second term in two cases. For 3 < r < ∞ let 1/q =1/r + 2/3 and use (3.8) and for r =∞ use (3.9)‖R0(−λ2)V g‖Lr .{ ‖V g‖Lq , 3 < r <∞‖V g‖L3/2−∩L3/2+ , r =∞.{‖V 〈x〉2‖Lm‖g‖L2−2 , 3 < r <∞‖V 〈x〉2‖L6−∩L6+‖g‖L2−2 , r =∞. ‖g‖L2−2 .Here we used that |V (x)| . 〈x〉−4, and with 1/q = 1/m + 1/2 we have(4 − 2)m > 3. Finally we appeal to Lemma 3.1.1 and use the fact thatLr ⊂ L2−2 to see‖R0(−λ2)V g‖Lr . ‖(1 +R0(−λ2)V )−1f‖L2−2 .1λ‖f‖L2−2 .1λ‖f‖Lrwhere we can remove the factor of 1/λ if our orthogonality condition issatisfied.In light of (3.10),1 +G0V : Lr ∩ V ⊥ → Lr ∩ (V ψ)⊥is bijective, and so we treat the operator (1 +G0V )−1 as acting(1 +G0V )−1 : Lr ∩ (V ψ)⊥ → Lr ∩ V ⊥,which is the meaning of the expression Q¯(1+G0V )−1P¯ involving the projec-tions P¯ and Q¯. That the range should be taken to be V ⊥ is a consequenceof estimate (3.14) below.To prove (3.11), expandR0(−λ2) = G0 − λG1 + λ2R˜,R˜ :=1λ2(R0(−λ2)−G0 + λG1)=1λ(e−λ|x| − 1 + λ|x|4piλ|x|)∗443.1. Construction of Solitary Wave Profilesand consider f ∈ (V ψ)⊥ ∩ Lr with 3 < r ≤ ∞. We first establish theestimates‖h‖Lq .{1, 1 < q <∞log(1/λ), q = 1, h := V R˜V ψ, (3.13)|〈V, (1 +R0(−λ2)V )−1f〉| .{λ, 3 < r <∞λ log(1/λ), r =∞}‖f‖Lr . (3.14)For the purpose of these estimates we may make the following replacements:V ψ → 〈x〉−5, V → 〈x〉−4, and R˜(x)→ min(|x|, 1/λ). To establish (3.13) wemust therefore estimate〈x〉−4∫R3min(|y|, 1/λ)〈y − x〉−5dy,and we proceed in two parts:• Take |y| ≤ 2|x|. Then〈x〉−4∫|y|≤2|x|min(|y|, 1/λ)〈y − x〉−5dy. 〈x〉−4 min(|x|, 1/λ)∫〈y − x〉−5dy. 〈x〉−4 min(|x|, 1/λ)and‖〈x〉−4 min(|x|, 1/λ)‖qLq.∫ 10rq+2dr +∫ 1/λ1r−3q+2dr +1λ∫ ∞1/λr−4q+2dr. 1 +{1, q > 1log(1/λ), q = 1}+ λ4(q−1).{1, q > 1log(1/λ), q = 1.• Take |y| ≥ 2|x|. Then〈x〉−4∫|y|≥2|x|min(|y|, 1/λ)〈y − x〉−5dy . 〈x〉−4∫|y|〈y〉−5dy. 〈x〉−4and‖〈x〉−4‖Lq . 1.453.1. Construction of Solitary Wave ProfilesWith (3.13) established we now prove (3.14). Let g = (1+R0(−λ2)V )−1fand observe0 =1λ〈V ψ, f〉=1λ〈V ψ, (1 +R0(−λ2)V )g〉=1λ〈(1 + V R0(−λ2))(V ψ), g〉=1λ〈(1 + V (G0 − λG1 + λ2R˜))(V ψ), g〉= 〈(−V G1 + λV R˜)(V ψ), g〉= − 1√4pi〈V, g〉+ λ〈h, g〉noting that (1 + V G0)(V ψ) = 0. Now|〈V, g〉| . λ‖h‖Lr′‖g‖Lr . λ{1, 3 < r <∞log(1/λ), r =∞}‖f‖Lrapplying (3.13).With (3.14) in place we finish the argument. For f ∈ Lr ∩ (V ψ)⊥ wewriteg = (1 +R0(−λ2)V )−1f and g0 = (1 +G0V )−1f.We have0 = (1 +R0(−λ2)V )g − (1 +G0V )g0and so(1 +G0V )(g − g0) = −RˆV gwhere Rˆ = R0(−λ2) − G0. The above also implies RˆV g ⊥ V ψ. We invertto seeg − g0 = −(1 +G0V )−1RˆV g + αψnoting that ψ ∈ ker(1 +G0V ). Take now inner product with V to seeα〈V, ψ〉 = 〈V, g〉463.1. Construction of Solitary Wave Profilesand so|α| . |〈V, g〉| .{λ, 3 < r <∞λ log(1/λ), r =∞}‖f‖Lrobserving (3.14). It remains to estimate (1 +G0V )−1RˆV g. We note thatRˆ =(e−λ|x| − 14pi|x|)∗and so for estimates we may replace Rˆ(x) with min(λ, 1/|x|). There followsby Young’s inequality‖(1 +G0V )−1RˆV g‖Lr . ‖RˆV g‖Lr. ‖min(λ, 1/|x|)‖Lr‖V g‖L1. ‖min(λ, 1/|x|)‖Lr‖g‖Lr. λ1−3/r‖f‖Lr .And so after putting everything together we obtain (3.11).We end this section by recording pointwise estimates of the nonlineartermsN(η) = (W + η)5 −W 5 − 5W 4η + ε (f(W + η)− f(W )) .Bound the first three terms as follows:|(W + η)5 −W 5 − 5W 4η| .W 3η2 + |η|5.For the other term we use the Fundamental Theorem of Calculus and As-sumption 1.7.1 to see|f(W + η)− f(W )| =∣∣∣∣∫ 10∂δf(W + δη)dδ∣∣∣∣=∣∣∣∣∫ 10f ′(W + δη)ηdδ∣∣∣∣. |η| sup0<δ<1(|W + δη|p1−1 + |W + δη|p2−1). |η| (W p1−1 + |η|p1−1 +W p2−1 + |η|p2−1). |η| (W p1−1 +W p2−1)+ |η|p1 + |η|p2473.1. Construction of Solitary Wave Profilesand so together we have|N(η)| .W 3η2 + |η|5 + ε|η| (W p1−1 +W p2−1)+ ε|η|p1 + ε|η|p2 . (3.15)Similarly|f(W + η1)− f(W + η2)|. |η1 − η2|(W p1−1 + |η1|p1−1 + |η2|p1−1 +W p2−1 + |η1|p2−1 + |η2|p2−1)and so|N(η1)−N(η2)| . |η1 − η2|(|η1|+ |η2|)W 3 + |η1 − η2|(|η1|4 + |η2|4)+ ε|η1 − η2|(W p1−1 +W p2−1)+ ε|η1 − η2|(|η1|p1−1 + |η2|p1−1 + |η1|p2−1 + |η2|p2−1) .(3.16)3.1.3 Solving for the FrequencyWe are now in a position to construct solutions to (1.11) and so proveTheorem 1.7.2. The proof proceeds in two steps. In the present section, wewill solve for λ in (3.3) for a given small η. Then in the following Section 3.4,we will treat λ as a function of η and solve (3.4). Both steps involve fixedpoint arguments.We begin by computing the inner product (3.3). Write0 = 〈R0(−λ2)V ψ,F〉 = 〈R0(−λ2)V ψ,−λ2W + εf(W ) +N(η)〉so thatλ · λ〈R0(−λ2)V ψ,W 〉 = ε〈R0(−λ2)V ψ, f(W )〉+ 〈R0(−λ2)V ψ,N(η)〉.(3.17)It is our intention to find a solution λ of (3.17) of the appropriate size. Thisis done in Lemma 3.1.6 but we first make some estimates on the leadingorder inner products appearing above.Lemma 3.1.5. We have the estimates〈R0(−λ2)V ψ, f(W )〉 = −〈ψ, f(W )〉+O(λδ1) (3.18)λ〈R0(−λ2)V ψ,W 〉 = 2√3pi +O(λ) (3.19)where δ1 is defined in the statement of Theorem Construction of Solitary Wave ProfilesProof. Firstly〈R0(−λ2)V ψ, f(W )〉 = 〈G0V ψ, f(W )〉+ 〈(R0(−λ2)−R0(0))V ψ, f(W )〉First note that since Hψ = 0 we have V ψ = −(−∆ψ) so〈G0V ψ, f(W )〉 = 〈−(−∆)−1(−∆ψ), f(W )〉 = −〈ψ, f(W )〉.Note that this inner product is finite. For the other term use the resolventidentity R0(−λ2)−R0(0) = −λ2R0(−λ2)R0(0) to see〈(R0(−λ2)−R0(0))V ψ, f(W )〉 = λ2〈R0(−λ2)ψ, f(W )〉.Observe now thatλ2|〈R0(−λ2)ψ, f(W )〉| ≤ λ2‖R0(−λ2)ψ‖Lr‖f(W )‖Lr∗where 1/r + 1/r∗ = 1. Choose an r∗ > 1 with 3/p1 < r∗ < 3/2. In this wayf(W ) ∈ Lr∗ observing Assumption 1.7.1. We now apply (3.7) with q = 3noting that 3 < r <∞. Henceλ2|〈R0(−λ2)ψ, f(W )〉| . λ2 · λ3(1/3−1/r)−2‖ψ‖L3w‖f(W )‖Lr∗. λ1−3/r.If p1 ≥ 3 we can take r as large as we like. Otherwise we must take 3 < r <3/(3−p1) and so 1−3/r can be made close to p1−2 (from below). We nowsee (3.18).Next on to (3.19). Note that this computation is taken from [44]. Firstwe isolate the troublesome part of W and writeW =√3|x| + W˜ .There is no problem with the second term since W˜ ∈ L6/5 and V ψ ∈ L6/5so we can use (3.8) with q = 6/5 and r = 6 to seeλ|〈R0(−λ2)V ψ, W˜ 〉| . λ‖R0(−λ2)V ψ‖L6‖W˜‖L6/5. λ‖V ψ‖L6/5‖W˜‖L6/5 (3.20). λ. (3.21)Set g := V ψ and concentrate onλ√3〈R0(−λ2)g, 1|x|〉=√6piλ〈gˆ(ξ)|ξ|2 + λ2 ,1|ξ|2〉493.1. Construction of Solitary Wave Profileswhere we work on the Fourier Transform side, using Plancherel’s theorem.So √6piλ〈gˆ(ξ)|ξ|2 + λ2 ,1|ξ|2〉=√6piλ gˆ(0)〈1|ξ|2 + λ2 ,1|ξ|2〉+√6piλ〈gˆ(ξ)− gˆ(0)|ξ|2 + λ2 ,1|ξ|2〉where the first term is the leading order. We invert the Fourier Transformand note that gˆ(0) = (2pi)−32∫g to see√6piλ gˆ(0)〈1|ξ|2 + λ2 ,1|ξ|2〉=√3(∫g)λ〈e−λ|x|4pi|x| ,1|x|〉=√3∫g = 2√3pi.We now must bound the remainder term. It is easy for the high frequencies∫|ξ|≥1|gˆ(ξ)− gˆ(0)||ξ|2(|ξ|2 + λ2)dξ . ‖gˆ‖L∞∫|ξ|≥1dξ|ξ|4 . ‖g‖L1 . 1.For the low frequencies note that since |x|g ∈ L1 we have that ∇gˆ is contin-uous and bounded. In light of this seth(ξ) := φ(ξ) (gˆ(ξ)− gˆ(0)−∇gˆ(0) · ξ)where φ is a smooth, compactly supported cutoff function with φ = 1 on|ξ| ≤ 1. Now since ∫|ξ|≤1ξ|ξ|2(|ξ|2 + λ2)dξ = 0we have ∫|ξ|≤1gˆ(ξ)− gˆ(0)|ξ|2(|ξ|2 + λ2)dξ =∫|ξ|≤1h(ξ)|ξ|2(|ξ|2 + λ2)dξand so bound this integral instead. If we recall the form of g we see |g| .〈x〉−5 and so (1+|x|1+α)g ∈ L1 for some α > 0. Therefore (1+|x|1+α)hˇ ∈ L1and noting also that ∇h(0) = 0 we see |∇h(ξ)| . min(1, |ξ|α). The Mean503.1. Construction of Solitary Wave ProfilesValue Theorem along with h(0) = 0 then gives |h(ξ)| . min(1, |ξ|1+α). Withthis bound established we consider two regions of the integral∫|ξ|≤λ|h(ξ)||ξ|2(|ξ|2 + λ2)dξ .∫|ξ|≤λ|ξ||ξ|2(|ξ|2 + λ2)dξ.∫|ζ|≤11|ζ|(|ζ|2 + 1)dζ . 1and∫λ≤|ξ|≤1|h(ξ)||ξ|2(|ξ|2 + λ2)dξ .∫λ≤|ξ|≤1|ξ|1+α|ξ|2(|ξ|2 + λ2)dξ. λα∫1≤|ζ|≤1/λ|ζ|α−1|ζ|2 + 1dζ . λα · λ−α . 1.Putting everything together gives (3.19).With the above estimates in hand we turn our attention to solving (3.17).Lemma 3.1.6. For any R > 0 there exists ε0 = ε0(R) > 0 such that for0 < ε ≤ ε0 and given a fixed η ∈ L∞ with ‖η‖L∞ ≤ Rε the equation (3.17)has a unique solution λ = λ(ε, η) satisfying ελ(1)/2 ≤ λ ≤ 3ελ(1)/2 whereλ(1) =−〈ΛW, f(W )〉6pi> 0. (3.22)Moreover, we have the expansionλ = λ(1)ε+ λ˜, λ˜ = O(ε1+δ1). (3.23)Remark 3.1.7. Writing the resolvent as (3.5), and thus the subsequentestimates (3.6)-(3.8), require λ > 0 and so it is essential that we have es-tablished λ(1) > 0. This is the source of the sign condition in Assumption1.7.1.Proof. We first estimate the remainder term. Take ελ(1)/2 ≤ λ ≤ 3ελ(1)/2and η with ‖η‖L∞ ≤ Rε. We establish the estimate|〈R0(−λ2)V ψ,N(η)〉| . ε1+δ1 . (3.24)We deal with each term in (3.15). Take j = 1, 2. We frequently apply (3.6),(3.8) and Ho¨lder:513.1. Construction of Solitary Wave Profiles• |〈R0(−λ2)V ψ,W 3η2〉| . ‖R0(−λ2)V ψ‖L6‖W 3η2‖L6/5. ‖V ψ‖L6/5‖η‖2L∞‖W 3‖L6/5. ε2• |〈R0(−λ2)V ψ, η5〉| . ‖R0(−λ2)V ψ‖L1‖η5‖L∞. λ−2‖V ψ‖L1‖η‖5L∞. ε3• ε|〈R0(−λ2)V ψ, ηpj 〉| . ε‖R0(−λ2)V ψ‖L1‖ηpj‖L∞. ελ−2‖V ψ‖L1‖η‖pjL∞. ε · εpj−2The term that remains requires two cases. First take pj > 3 thenε|〈R0(−λ2)V ψ, ηW pj−1〉| . ε‖R0(−λ2)V ψ‖Lr‖ηW pj−1‖Lr∗. ε‖V ψ‖Lq‖η‖L∞‖W pj−1‖Lr∗. ε2where we have used (3.8) for some r∗ < 3/2 and r > 3. Now if instead2 < pj ≤ 3 we use (3.6) with r∗ = (3/(pj − 1))+ so 1− 1/r = ((pj − 1)/3)−andε|〈R0(−λ2)V ψ, ηW pj−1〉| . ε‖R0(−λ2)V ψ‖Lr‖ηW pj−1‖Lr∗. ελ3(1−1/r)−2‖V ψ‖L1‖η‖L∞‖W pj−1‖Lr∗. ε · ε(pj−2)−and so we establish (3.24).With the estimates (3.18), (3.19), (3.24) in hand we show that a solutionto (3.17) of the desired size exists. For this write (3.17) as a fixed pointproblemλ = H(λ) := ε〈R0(−λ2)V ψ, f(W )〉+ 〈R0(−λ2)V ψ,N(η)〉λ〈R0(−λ2)V ψ,W 〉 (3.25)with the intention of applying Banach Fixed Point Theorem. We show thatfor a fixed η with ‖η‖L∞ . ε the function H maps the interval ελ(1)/2 ≤λ ≤ 3ελ(1)/2 to itself and that H is a contraction.First note that −〈ψ, f(W )〉 > 0 by Assumption 1.7.1 and so after ob-serving (3.18), (3.19), (3.24) we see that H(λ) > 0. Furthermore for ε small523.1. Construction of Solitary Wave Profilesenough we have that ελ(1)/2 ≤ H(λ) ≤ 3ελ(1)/2 and so H maps this intervalto itself.We next show that H is a contraction. Take ελ(1)/2 ≤ λ1, λ2 ≤ 3ελ(1)/2and again keep η fixed with ‖η‖L∞ ≤ Rε. WriteH(λ) = a(λ) + b(λ)c(λ)so that|H(λ1)−H(λ2)| ≤ |a1||c2 − c1|+ |a1 − a2||c1|+ |b1||c2 − c1|+ |b1 − b2||c1||c1c2|. |a1 − a2|+ |b1 − b2|+ ε|c1 − c2|using (3.18), (3.19), (3.24). We treat each piece in turn.First|a1 − a2| = ε|〈(R0(−λ21)−R0(−λ22))V ψ, f(W )〉|= ε|λ21 − λ22||〈R0(−λ21)R0(−λ22)V ψ, f(W )〉|by the resolvent identity. Continuing we see|a1 − a2| . ε2|λ1 − λ2|‖R0(−λ21)R0(−λ22)V ψ‖Lr‖f(W )‖Lr∗where 1/r + 1/r∗ = 1. Note that by Assumption 1.7.1 we have f(W ) ∈ Lr∗for some 1 < r∗ < 3/2 so 3 < r <∞. Applying now (3.8) we get|a1 − a2| . ε2|λ1 − λ2|‖R0(−λ22)V ψ‖Lqwith 3(1/q − 1/r) = 2 so 1 < q < 3/2. Now apply the bound (3.6)|a1 − a2| . ε2|λ1 − λ2|λ3(1−1/q)−2‖V ψ‖L1. ε3(1−1/q)|λ1 − λ2|and note that 3(1− 1/q) > 0.Next consider|b1 − b2| = |〈R0(−λ21)V ψ,N(η)〉 − 〈R0(−λ22)V ψ,N(η)〉|.Proceeding as in the previous argument and using (3.6) we see|b1 − b2| . ε|λ1 − λ2|‖R0(−λ21)R0(−λ22)V ψ‖Lr‖N(η)‖Lr∗. ε|λ1 − λ2|λ−21 ‖R0(−λ22)V ψ‖Lr‖N(η)‖Lr∗533.1. Construction of Solitary Wave Profilesfor 1/r + 1/r∗ = 1. We can estimate this term (using different r and r∗for different portions of N(η)) using the computations leading to (3.24) toachieve|b1 − b1| . ε−1 · ε1+δ1 |λ1 − λ2| = εδ1 |λ1 − λ2|.Lastly considerε|c1 − c2| = ε|λ1〈R0(−λ21)V ψ,W 〉 − λ2〈R0(−λ22)V ψ,W 〉|.Again we write W =√3/|x|+W˜ where W˜ ∈ L6/5. The second term is easy.We computeε|λ1〈R0(−λ21)V ψ, W˜ 〉 − λ2〈R0(−λ22)V ψ, W˜ 〉|. ε|λ1 − λ2||〈R0(−λ21)V ψ, W˜ 〉|+ ε3|λ1 − λ2||〈R0(−λ21)R0(−λ22)V ψ, W˜ 〉|. ε|λ1 − λ2|+ ε3λ−21 λ3(1−1/6)−22 |λ1 − λ2|‖V ψ‖L1‖W˜‖L6/5. ε|λ1 − λ2|where we have used (3.21) once and (3.6) twice. For the harder term wefollow the computations which establish (3.19) and so work on the FourierTransform sideελ1〈R0(−λ21)V ψ, 1/|x|〉 − ελ2〈R0(−λ22)V ψ, 1/|x|〉= Cελ1〈gˆ(ξ)|ξ|2 + λ21,1|ξ|2〉− Cελ2〈gˆ(ξ)|ξ|2 + λ22,1|ξ|2〉= Cελ1〈gˆ(ξ)− gˆ(0)|ξ|2 + λ21,1|ξ|2〉− Cελ2〈gˆ(ξ)− gˆ(0)|ξ|2 + λ22,1|ξ|2〉= Cε(λ1 − λ2)〈gˆ(ξ)− gˆ(0)|ξ|2 + λ21,1|ξ|2〉+ Cελ2〈(gˆ(ξ)− gˆ(0))(1|ξ|2 + λ21− 1|ξ|2 + λ22),1|ξ|2〉where we have used the fact thatλ1〈gˆ(0)|ξ|2 + λ21,1|ξ|2〉= λ2〈gˆ(0)|ξ|2 + λ22,1|ξ|2〉.Continuing as in the computations used to establish (3.19) , we boundε|λ1 − λ2|∣∣∣∣〈 gˆ(ξ)− gˆ(0)|ξ|2 + λ21 , 1|ξ|2〉∣∣∣∣ . ε|λ1 − λ2|543.1. Construction of Solitary Wave Profilesandελ2∣∣∣∣〈(gˆ(ξ)− gˆ(0))( 1|ξ|2 + λ21 − 1|ξ|2 + λ22),1|ξ|2〉∣∣∣∣. ελ2(λ1 + λ2)|λ1 − λ2|∫dξ|ξ|(|ξ|2 + λ21)(|ξ|2 + λ22). ε|λ1 − λ2|∫dζ|ζ|(|ζ|2 + 1)(|ζ|2 + λ22/λ21). ε|λ1 − λ2|.In this way we finally haveε|c1 − c2| ≤ ε|λ1 − λ2|.So, putting everything together we see that by taking ε sufficiently small,|H(λ1)−H(λ2)| < κ|λ1 − λ2|for some 0 < κ < 1, and hence H is a contraction. Therefore (3.25) has aunique fixed point of the desired size.To find the leading order λ(1) let λ take the form in (3.23), substituteto (3.17) use estimates (3.18), (3.19), (3.24) and ignore higher order terms.An inspection of the higher order terms gives the order of λ˜.In this way we now think of λ as a function of η. We will also need thefollowing Lipshitz condition for what follows in Lemma 3.1.9.Lemma 3.1.8. The λ generated via Lemma 3.1.6 is Lipshitz continuous inη in the sense that|λ1 − λ2| . εδ1‖η1 − η2‖L∞ .Proof. Take η1 and η2 with ‖η1‖L∞ , ‖η2‖L∞ ≤ Rε. Let η1 and η2 give rise toλ1 and λ2 respectively through Lemma 3.1.6. Consider now the difference|λ1 − λ2| =∣∣∣∣ε〈R0(−λ21)V ψ, f(W )〉+ 〈R0(−λ21)V ψ,N(η1)〉λ1〈R0(−λ21)V ψ,W 〉− ε〈R0(−λ22)V ψ, f(W )〉+ 〈R0(−λ22)V ψ,N(η2)〉λ2〈R0(−λ22)V ψ,W 〉∣∣∣∣=:∣∣∣∣a(λ1) + b(λ1, η1)c(λ1) − a(λ2) + b(λ2, η2)c(λ2)∣∣∣∣553.1. Construction of Solitary Wave Profilesobserving (3.25). Now we estimate|λ1 − λ2| ≤∣∣∣∣b(λ1, η1)− b(λ1, η2)c(λ1)∣∣∣∣+ ∣∣∣∣a(λ1) + b(λ1, η2)c(λ1) − a(λ2) + b(λ2, η2)c(λ2)∣∣∣∣≤C|〈R0(−λ21)V ψ,N(η1)−N(η2)〉|+ κ|λ1 − λ2|for some 0 < κ < 1. The second term has been estimated using the com-putations of Lemma 3.1.6 and taking ε small enough. Now we estimatethe first. Observing the terms in (3.16) we use the same procedure thatestablished (3.24) to obtain|〈R0(−λ21)V ψ,N(η1)−N(η2)〉| . εδ1‖η1 − η2‖L∞ .So together we now see(1− κ)|λ1 − λ2| . εδ1‖η1 − η2‖L∞which gives the desired result.3.1.4 Solving for the CorrectionWe next solve (3.4), given that (3.3) holds. Recall the formulation of (3.4)as the fixed-point equationη = G(η, ε) = (H + λ2)−1Fwhere in light of Lemma 3.1.6, we take λ = λ(ε, η) and F = F(ε, λ(ε, η), η)so that (3.3) holds.Lemma 3.1.9. There exists R0 > 0 such that for any R ≥ R0, there isε1 = ε1(R) > 0 such that for each 0 < ε ≤ ε1, there exists a unique solutionη ∈ L∞ to (3.4) with ‖η‖L∞ ≤ Rε. Moreover, we have the expansionη = εQ¯(1 +G0V )−1P¯(G0f(W )− λ(1)√3λR0(−λ2)|x|−1)+OL∞(ε1+δ1)where P¯ and Q¯ are given in (3.12).Proof. We proceed by means of Banach Fixed Point Theorem. We showthat G(η) maps a ball to itself and is a contraction. In this way we establisha solution to η = G(η, ε, λ(ε, η)) in (3.4).Let R > 0 (to be chosen) and take ε < ε0(R) as in Lemma 3.1.6. In thisway given η ∈ L∞ with ‖η‖L∞ ≤ Rε we can generateλ = λ(ε, η) = λ(1)ε+ o(ε).563.1. Construction of Solitary Wave ProfilesWe aim to take ε smaller still in order to run fixed point in the L∞ ball ofradius Rε.Consider‖G‖L∞ = ‖(1 +R0(−λ2)V )−1R0(−λ2)F‖L∞. ‖R0(−λ2)F‖L∞in light of Lemma 3.1.4 and since we have chosen λ to satisfy (3.3). Contin-uing with‖G‖L∞ . ‖R0(−λ2)(−λ2W + εf(W ) +N(η)) ‖L∞we treat each term separately. For the first term it is sufficient to replaceW with 1/|x| (otherwise we simply apply (3.9))λ2‖R0(−λ2)W‖L∞ . λ∥∥∥∥λR0(−λ2) 1|x|∥∥∥∥L∞. λ∥∥∥∥∥λ∫e−λ|y||y|1|x− y|dy∥∥∥∥∥L∞. λ∥∥∥∥∥∫e−|z||z|1|λx− z|dz∥∥∥∥∥L∞. λ∥∥∥∥∥(e−|x||x| ∗1|x|)(λx)∥∥∥∥∥L∞. λ . ε.Now for the second term use (3.9)ε‖R0(−λ2)f(W )‖L∞ . ε‖f(W )‖L3/2−∩L3/2+ . ε.And for the higher order terms we employ (3.6) and (3.9)• ‖R0(−λ2)(W 3η2)‖L∞ . ‖W 3η2‖L3/2−∩L3/2+. ‖W 3‖L3/2−∩L3/2+‖η‖2L∞. R2ε2• ‖R0(−λ2)η5‖L∞ . λ−2‖η5‖L∞ . λ−2‖η‖5L∞ . R5ε3• ε‖R0(−λ2)ηpj‖L∞ . ελ−2‖ηpj‖L∞. ε−1‖η‖pjL∞. Rpjεpj−1573.1. Construction of Solitary Wave Profilesfor j = 1, 2. The remaining remainder term again requires two cases. Forpj > 3 we use (3.9) to seeε∥∥R0(−λ2) (ηW pj−1)∥∥L∞ . ε‖ηW pj−1‖L3/2−∩L3/2+. ε‖η‖L∞‖W pj−1‖L3/2−∩L3/2+. Rε2and for 2 < pj ≤ 3 we apply (3.6)ε∥∥R0(−λ2) (ηW pj−1)∥∥L∞ . ελ(pj−1)−−2‖η‖L∞‖W pj−1‖L3/(pj−1)+. Rε1+(pj−2)−Collecting the above yields‖G‖L∞ ≤ Cε(1 +R2ε+R5ε2 +Rp1εp1−2 +Rp2εp2−2 +Rε+Rε(p1−2)−)(3.26)and so taking R0 = 2C, R ≥ R0, and then ε small enough so that Rε +R4ε2 +Rp1−1εp1−2 +Rp2−1εp2−2 + ε+ ε(p1−2)− ≤ 12C , we arrive at‖G‖L∞ ≤ Rε.Hence G maps the ball of radius Rε in L∞ to itself.Now we show that G is a contraction. Take η1 and η2 and let them giverise to λ1 and λ2 respectively. Again ‖ηj‖L∞ ≤ Rε and denote F(ηj) by Fj ,j = 1, 2. Consider‖G(η1, ε)− G(η2, ε)‖L∞= ‖(1 +R0(−λ21)V )−1R0(−λ21)F1 − (1 +R0(−λ22)V )−1R0(−λ22)F2‖L∞≤ ‖(1 +R0(−λ21)V )−1(R0(−λ21)F1 −R0(−λ22)F2) ‖L∞+ ‖ ((1 +R0(−λ21)V )−1 − (1 +R0(−λ22)V )−1)R0(−λ22)F2‖L∞≤ ‖R0(−λ21)F1 −R0(−λ22)F2‖L∞+ ‖ ((1 +R0(−λ21)V )−1 − (1 +R0(−λ22)V )−1)R0(−λ22)F2‖L∞≤ ‖R0(−λ21) (F1 −F2) ‖L∞ + ‖(R0(−λ21)−R0(−λ22))F2‖L∞+ ‖ ((1 +R0(−λ21)V )−1 − (1 +R0(−λ22)V )−1)R0(−λ22)F2‖L∞=: I + II + IIIwhere we have applied Lemma 3.1.4, observing the orthogonality condition.We treat each part in turn.583.1. Construction of Solitary Wave ProfilesStart with I. This computation is similar to those previous. We alsoapply Lemma 3.1.8:‖R0(−λ21) (F1 −F2) ‖L∞ = ‖R0(−λ21)((λ22 − λ21)W +N(η1)−N(η2)) ‖L∞. |λ1 − λ2|+ εδ1‖η1 − η2‖L∞. εδ1‖η1 − η2‖L∞ .Part II is also similar to previous computations:‖ (R0(−λ21)−R0(−λ22))F2‖L∞ = |λ21 − λ22|‖R0(−λ21)R0(−λ22)F2‖L∞. |λ1 + λ2||λ1 − λ2|λ−21 ‖R0(−λ22)F2‖L∞. ε1 · ε−2 · ε|λ1 − λ2|. |λ1 − λ2|. εδ1‖η1 − η2‖L∞ .Part III is the hardest. First we find a common denominator(1 +R0(−λ21)V )−1 − (1 +R0(−λ22)V )−1= (1 +R0(−λ21)V )−1(1 +R0(−λ22)V )(1 +R0(−λ22)V )−1− (1 +R0(−λ21)V )−1(1 +R0(−λ21)V )(1 +R0(−λ22)V )−1= (1 +R0(−λ21)V )−1(R0(−λ22)V −R0(−λ21)V)(1 +R0(−λ22)V )−1so that((1 +R0(−λ21)V )−1 − (1 +R0(−λ22)V )−1)R0(−λ22)F2 =(1 +R0(−λ21)V )−1(R0(−λ22)V −R0(−λ21)V )(1 +R0(−λ22)V )−1R0(−λ22)F2= (1 +R0(−λ21)V )−1(R0(−λ22)V −R0(−λ21)V)G(η2).NowIII = ‖(1 +R0(−λ21)V )−1(R0(−λ22)V −R0(−λ21)V)G(η2)‖L∞and here we just suffer the loss of one λ (Lemma 3.1.4) to achieveIII . λ−11 ‖(R0(−λ22)V −R0(−λ21)V)G(η2)‖L∞. λ−11 |λ22 − λ21|‖R0(−λ22)R0(−λ21)V G(η2)‖L∞. λ−11 |λ2 + λ1||λ2 − λ1|λ−1/22 ‖R0(−λ21)V G(η2)‖L2. ε−1 · ε1|λ2 − λ1|λ−1/22 λ−1/21 ‖V G(η2)‖L1. ε−1|λ2 − λ1|‖V ‖L1‖G(η2)‖L∞593.1. Construction of Solitary Wave Profilesand using Lemma 3.1.8 and (3.26) we seeIII . |λ1 − λ2| . εδ1‖η1 − η2‖L∞ .Hence, by taking ε smaller still if needed, we have‖G(η1, ε)− G(η2, ε)‖L∞ ≤ κ‖η1 − η2‖L∞for some 0 < κ < 1 and so G is a contraction. Therefore, invoking theBanach fixed-point theorem, we have established the existence of a uniqueη, with ‖η‖L∞ ≤ Rε, satisfying (3.4).To see the leading order observe the order of the terms appearing in theprevious computations as well as the following. First if p1 ≥ 3 thenε‖ (R0(−λ2)−G0) f(W )‖L∞ . ελ2‖R0(−λ2)G0f(W )‖L∞. ελ2 · λ−1−‖G0f(W )‖L3+. ελ1−‖f(W )‖L1+. ε2−and if instead 2 < p1 < 3 then take 3/q = (p1 − 2)− andε‖ (R0(−λ2)−G0) f(W )‖L∞ . ελ2‖R0(−λ2)G0f(W )‖L∞. ελ2 · λ3/q−2‖G0f(W )‖Lq. ελ3/q‖f(W )‖L(3/p1)+. ε1+(p1−2)− .The lemma is now proved.With the existence of η established we can improve the space in whichη lives.Lemma 3.1.10. The η established in Lemma 3.1.9 is in Lr ∩ H˙1 for any3 < r ≤ ∞. The function η also enjoys the bounds‖η‖Lr . ε1−3/r‖η‖H˙1 . ε1/2for all 3 < r ≤ ∞. Furthermore we have the expansionη = Q¯(1 +G0V )−1P¯R0(−λ2)(−λ2√3|x|−1) + η˜603.1. Construction of Solitary Wave Profileswith‖η˜‖Lr .max{{ε1−, if 2 < p1 < 3 and r = 3/(p1 − 2)ε, else}, εp1−2+1−3/r, ε2(1−3/r)}for 3 < r <∞ and where P¯ and Q¯ are given in (3.12).Proof. The computations which produce (3.26) are sufficient to establish theresult with r =∞. Take 3 < r <∞ and consider:‖η‖Lr . λ2‖R0(−λ2)W‖Lr + ε‖R0(−λ2)f(W )‖Lr + ‖R0(−λ2)N(η)‖Lr .For the first term use (3.7)λ2‖R0(−λ2)W‖Lr . λ2 · λ3(1/3−1/r)−2‖W‖L3w . ε1−3/rto see the leading order contribution.While the second term contributed to the leading order in Lemma 3.1.9it is inferior to the first term when measured in Lr. We do however needseveral cases. Suppose that 3 ≤ p1 < 5 or r > 3/(p1 − 2) and apply (3.8)with 1/q = 1/r + 2/3ε‖R0(−λ2)f(W )‖Lr . ε‖f(W )‖Lq . ε.Note that under these conditions f(W ) ∈ Lq. Now suppose that 2 < p1 < 3and r = 3/(p1 − 2) and apply (3.6) with q = (3/p1)+ε‖R0(−λ2)f(W )‖Lr . ελ3(1/q−(p1−2)/3)−2‖f(W )‖Lq . ε1− .And if 2 < p1 < 3 and 3 < r < 3/(p1 − 2) apply (3.7) with q = 3/p1ε‖R0(−λ2)f(W )‖Lr . ελ3(p1/3−1/r)−2‖f(W )‖Lqw . ε1−3/r+p1−2.And thirdly the remaining terms. First use (3.8) where 1/q = 1/r+ 2/3to see‖R0(−λ2)(W 3η2)‖Lr . ‖W 3η2‖Lq . ‖W 3‖L3/2‖η‖Lr‖η‖L∞ . ε‖η‖Lrand now use (3.6) with 1/q = 1/r to obtain‖R0(−λ2)η5‖Lr . λ−2‖η5‖Lr . λ−2‖η‖Lr‖η‖4L∞ . ε2‖η‖Lr .613.1. Construction of Solitary Wave Profilesand similarly for j = 1, 2ε‖R0(−λ2)ηpj‖Lr . ελ−2‖ηpj‖Lr . ε−1‖η‖pj−1L∞ ‖η‖Lr . εpj−2‖η‖Lrnoting that p2 − 2 ≥ p1 − 2 > 0. For the last remainder term we have twocases. If pj > 3 then use (3.8) with 1/q = 1/r + 2/3ε‖R0(−λ2)(ηW pj−1) ‖Lr . ε‖ηW pj−1‖Lq . ε‖η‖Lr‖W pj−1‖L3/2 . ε‖η‖Lr .If instead 2 < pj ≤ 3 then we need (3.7) with 1/q = (pj − 1)/3 so thatε‖R0(−λ2)(ηW pj−1) ‖Lr . ελ3(1/q−1/r)−2‖ηW pj−1‖Lqw. ελp1−1−3/r−2‖η‖L∞‖W pj−1‖Lqw. εpj−2ε1−3/rSo together we have‖η‖Lr ≤ Cε1−3/r + κ‖η‖Lrwhere κ may be chosen sufficiently small to yield the desired Lr bound for3 < r < ∞. An inspection of the higher order terms gives the size of η˜.We also must note Lemma 3.1.4. There are several competing terms whichdetermine the size of η˜ depending on p1 and r.On to the H˙1 norm. We need the identityη = (1 +R0(−λ2)V )−1R0(−λ2)F= R0(−λ2)F −R0(−λ2)V (1 +R0(−λ2)V )−1R0(−λ2)F= R0(−λ2)F −R0(−λ2)V ηso we have two parts‖η‖H˙1 ≤ ‖R0(−λ2)F‖H˙1 + ‖R0(−λ2)V η‖H˙1 .For the first‖R0(−λ2)F‖H˙1 . λ2‖R0(−λ2)W‖H˙1 + ε‖R0(−λ2)f(W + η)‖H˙1+ ‖R0(−λ2)(W 3η2 +W 2η3 +Wη4 + η5) ‖H˙1andλ2‖R0(−λ2)W‖H˙1 . λ2‖R0(−λ2)∇W‖L2. λ2 · λ1/2−2‖∇W‖L3/2w. ε1/2623.1. Construction of Solitary Wave Profilesandε‖R0(−λ2)f(W + η)‖H˙1 . ε‖R0(−λ2)f ′(W + η)(∇W +∇η)‖L2. ελ−1/2‖f ′(W + η)∇W‖L1+ ελ1−‖f ′(W + η)∇η‖L6/5−. ε1/2‖f ′(W + η)‖L3−‖∇W‖L3/2++ ε0+‖f ′(W + η)‖L3−‖∇η‖L2. ε1/2 + κ‖η‖H˙1with κ small and‖R0(−λ2)(W 3η2 +W 2η3 +Wη4 + η5) ‖H˙1. ‖R0(−λ2)η(∇Wf1 +∇ηf2)‖L2where f1 and f2 are in L2 so‖R0(−λ2)(W 3η2 +W 2η3 +Wη4 + η5) ‖H˙1. λ−1/2‖η‖L∞ (‖∇W‖L2‖f1‖L2 + ‖∇η‖L2‖f2‖L2). ε1/2 + κ‖η‖H˙1 .For the second‖R0(−λ2)V η‖H˙1 =∥∥∥∥∥(∇e−λ|x||x|)∗ (V η)∥∥∥∥∥L2=∥∥(λ2g(λx)) ∗ (V η)∥∥L2where g ∈ L3/2w . So using weak Young’s we obtain‖R0(−λ2)V η‖H˙1 . λ2‖g(λx)‖L3/2w ‖V η‖L6/5. λ2 · λ−2‖V ‖L3/2‖η‖L6. ‖η‖L6. ε1/2.So putting everything together gives‖η‖H˙1 ≤ C(ε1/2 + κ‖η‖H˙1)which gives the desired bound by taking κ sufficiently small.633.1. Construction of Solitary Wave ProfilesCombining Lemmas 3.1.6, 3.1.9, 3.1.10 and Remark 1.7.6 completes theproof of Theorem 1.7.2.At this point we demonstrate the following monotonicity result whichwill be used in Section 3.2.Lemma 3.1.11. Suppose that f(W ) = W p with 3 < p < 5. Take ε1 and ε2with 0 < ε1 < ε2 < ε0. Let ε1 give rise to λ1 and η1 and let ε2 give rise toλ2 and η2 via Theorem 1.7.2. We have|(λ2 − λ1)− λ(1)(ε2 − ε1)| . o(1)|ε2 − ε1|. (3.27)Proof. We first establish the estimate|λ2 − λ1| ≤(λ(1) + o(1))|ε2 − ε1| (3.28)We write, as in Lemma 3.1.6 and Lemma 3.1.8λ2 − λ1 = a(ε2, λ2) + b(ε2, λ2, η2)c(λ2)− a(ε1, λ1) + b(ε1, λ1, η1)c(λ1)=a(ε2, λ2)− a(ε1, λ2) + b(ε2, λ2, η2)− b(ε1, λ2, η2)c(λ2)+a(ε1, λ2) + b(ε1, λ2, η2)c(λ2)− a(ε1, λ1) + b(ε1, λ1, η1)c(λ1).The second line, containing only ε1 and not ε2, has been dealt with in theproof of Lemma 3.1.8 and so there follows|λ2 − λ1| ≤∣∣∣∣a(ε2, λ2)− a(ε1, λ2) + b(ε2, λ2, η2)− b(ε1, λ2, η2)c(λ2)∣∣∣∣+ o(1)‖η2 − η1‖L∞ + o(1)|λ2 − λ1|≤ |ε2 − ε1|(λ(1) + o(1))+ o(1)‖η2 − η1‖L∞ + o(1)|λ2 − λ1|.For the η’s we estimate‖η2 − η1‖L∞ . o(1)‖η2 − η1‖L∞ + |λ2 − λ1|+ |ε2 − ε1|appealing to Lemma 3.1.9. So putting everything together we have|λ2 − λ1| ≤(λ(1) + o(1))|ε2 − ε1|establishing (3.28).643.2. Variational CharacterizationNow we proceed to the more refined (3.27). Observing the computationsleading to (3.28) we have|λ2 − λ1 − (ε2 − ε1)λ(1)| ≤∣∣∣∣a(ε2, λ2)− a(ε1, λ2)c(λ2) − (ε2 − ε1)λ(1)∣∣∣∣+o(1)‖η2 − η1‖L∞ + o(1)|λ2 − λ1|.By (3.28) the last two terms are of the correct size and so we focus on thefirst. We have∣∣∣∣a(ε2, λ2)− a(ε1, λ2)c(λ2) − (ε2 − ε1)λ(1)∣∣∣∣=∣∣∣∣(ε2 − ε1)( 〈R0(−λ22)V ψ,W p〉λ2〈R0(−λ22)V ψ,W 〉 − λ(1))∣∣∣∣= o(1)|ε2 − ε1|noting (3.18) and (3.19). And so, putting everything together we achieve|λ2 − λ1 − (ε2 − ε1)λ(1)| . o(1)|ε2 − ε1|as desired.3.2 Variational CharacterizationIt is not clear from the construction that the solution Q is in any sense aground state solution. It is also not clear that the solution is positive. Inthis section we first establish the existence of a ground state solution; onethat minimizes the action subject to a constraint. We then demonstratethat this minimizer must be our constructed solution. In this way we proveTheorem 1.7.7.In this section we restrict our nonlinearity and take only f(Q) = |Q|p−1Qwith 3 < p < 5. Then the action isSε,ω(u) = 12‖∇u‖2L2 −16‖u‖6L6 −εp+ 1‖u‖p+1Lp+1+ω2‖u‖2L2 . (3.29)We are interested in the constrained minimization problemmε,ω := inf{Sε,ω(u) | u ∈ H1(R3) \ {0}, Kε(u) = 0} (3.30)whereKε(u) = ddµSε,ω(Tµu)∣∣∣∣µ=1= ‖∇u‖2L2 − ‖u‖6L6 −3(p− 1)2(p+ 1)ε‖u‖p+1Lp+1653.2. Variational Characterizationand (Tµu)(x) = µ3/2u(µx) is the L2 scaling operator. Note that for Qε =W + η as constructed in Theorem 1.7.2 we have Kε(Qε) = 0 since anysolution to (1.11) will satisfy Kε(Q) = 0.Before addressing the minimization problem we investigate the impli-cations of our generated solution Qε with specified ε and correspondingω = ω(ε). In particular there is a scaling that generates for us additionalsolutions to the equation−∆Q−Q5 − ε|Q|p−1Q+ ωQ = 0 (3.31)with 3 < p < 5.Remark 3.2.1. For any 0 < ε˜ ≤ ε0, we have solutions to (3.31) given byQµ = µ1/2Qε˜(µ·)with ε = µ(5−p)/2ε˜ and ω = µ2ω(ε˜). So for any ε > 0, we obtain the familyof solutions{ Qµ | µ =(εε˜) 25−p, ε˜ ∈ (0, ε0] }withω =(εε˜) 45−pω(ε˜) ∈[(ε0ε˜0) 45−pω(ε˜0), ∞)since as ε˜ ↓ 0, ( εε˜) 45−p ω(ε˜) ∼ ε˜ 2(3−p)5−p →∞.We now address the minimization problem by first addressing the exis-tence of a minimizer.Lemma 3.2.2. Take 3 < p < 5. Let Q = Qε solving (3.31) with ω = ω(ε)be as constructed in Theorem 1.7.2. There exists ε0 > 0 such that for 0 <ε ≤ ε0 we haveSε,ω(ε)(Qε) <13‖W‖6L6 = S0,0(W ).It follows, see Proposition 2.1 of [1], which is in turn based on the ear-lier [12], that the variational problem (3.30) with ω = ω(ε) admits a non-negative, radially-symmetric minimizer, which moreover solves (3.31).663.2. Variational CharacterizationProof. We compute directly, ignoring higher order contributions. Using(1.21) we write the action asSε,ω(Q) = 13∫Q6 +p− 12(p+ 1)ε∫|Q|p+1=13∫(W + η)6 +p− 12(p+ 1)ε∫|W + η|p+1.Rearranging we haveSε,ω(Q)− 13∫W 6 = 2∫W 5η +p− 12(p+ 1)ε∫W p+1 +O(ε2)where the higher order terms are controlled for 3 < p < 5:• ‖W 4η2‖L1 . ‖W 4‖L1‖η‖2L∞ . ε2• ‖η6‖L1 . ‖η‖6L6 . ε3• ε‖W pη‖L1 . ε‖W p‖L1‖η‖L∞ . ε2• ε‖ηp+1‖L1 . ε‖η‖p+1Lp+1 . εp−1.We now compute2∫W 5η = 2〈W 5, (H + λ2)−1(εW p − λ2W +N(η))〉= 2〈W 5, (1 +R0(−λ2)V )−1P¯R0(−λ2)(εW p − λ2W +N(η))〉where we have inserted the definition of η from (3.4) and so identify the twoleading order terms. There is no problem to also insert the projection P¯from (3.12) since we have the orthogonality condition (3.3) by the way wedefined ε, λ, η.We approximate in turn writing only R0 for R0(−λ2). In what followswe use the operators (1 + V G0)−1 and (1 + V R0)−1. The former as acts onthe spaces(1 + V G0)−1 : L1 ∩ (ΛW )⊥ → L1 ∩ (1)⊥and the later has the expansion(1 + V R0)−1 =1λ〈ΛW, ·〉V ΛW +O(1)673.2. Variational Characterizationin L1. We record here also the adjoint of P¯ :P¯ ∗ = 1− P ∗, P ∗ = 〈ΛW, ·〉∫V (ΛW )2V ΛW.To estimate the first term write2ε〈W 5, (1 +R0V )−1P¯R0W p〉 = 2ε〈(1 + V R0)−1W 5, P¯R0W p〉= 2ε〈(1 + V G0)−1W 5, P¯R0W p〉+O(ε2).The error is controlled with a resolvent identity:ε∣∣〈((1 + V R0)−1 − (1 + V G0)−1)W 5, P¯R0W p〉∣∣= ε∣∣〈(1 + V R0)−1V (G0 −R0)(1 + V G0)−1W 5, P¯R0W p〉∣∣= ε∣∣〈P¯ ∗(1 + V R0)−1V (G0 −R0) (−W 5/4 + V ΛW/2) , R0W p〉∣∣. ε∥∥P¯ ∗(1 + V R0)−1V (G0 −R0) (−W 5/4 + V ΛW/2)∥∥L1 ‖R0W p‖L∞. ε∥∥V R¯ (−W 5/4 + V ΛW/2)∥∥L1‖W p‖L3/2−∩L3/2+. ελ. ε2where we have substituted G0 − R0 = λG1 + R¯, note that G1(−W 5/4 +V ΛW/2) = 0 since (−W 5/4 + V ΛW/2) ⊥ 1, and have computed∥∥V R¯ (−W 5/4 + V ΛW/2)∥∥L1.∫〈x〉−1dx∫λ|λy|〈λy〉〈x− y〉−5dy . λ.Continuing, we have2ε〈W 5, (1 +R0V )−1P¯R0W p〉 = 2ε〈P¯ ∗(1 + V G0)−1W 5, R0W p〉+O(ε2)= −12ε〈W 5, R0W p〉+O(ε2)= −12ε〈R0W 5,W p〉+O(ε2)= −12ε〈G0W 5,W p〉+O(ε2)= −12ε〈W,W p〉+O(ε2)= −12ε∫W p+1 +O(ε2)683.2. Variational Characterizationwhere the other error term is bounded:ε∣∣〈(R0 −G0)W 5,W p〉∣∣ . ελ2 ∣∣〈R0G0W 5,W p〉∣∣ . ελ2 |〈R0W,W p〉| . ε2observing the computations that produce (3.19).For the second term we proceed in a similar manner−2λ2〈W 5, (1 +R0V )−1P¯R0W 〉 = 12λ2〈W 5, R0W 〉+O(ε2−)=12λ√3∫W 5 +O(ε2−)= 6piλ+O(ε2−)= −ε〈ΛW,W p〉+O(ε2−)= ε(3p+ 1− 12)∫W p+1 +O(ε2−)again referring to (3.19) and also Remark 1.7.4. The error term coming fromthe difference of the resolvents is similar. Noteλ2∣∣〈((1 + V R0)−1 − (1 + V G0)−1)W 5, P¯R0W〉∣∣. λ2∥∥P¯ ∗(1 + V R0)−1V (G0 −R0) (−W 5/4 + V ΛW/2)∥∥L1 ‖R0W‖L∞. λ3 ‖R0W‖L∞. λ3λ−1−‖W‖L3+. λ2−. ε2− .The term coming from N(η) is controlled similarly, and so, all togetherwe haveSε,ω(Q)− 13∫W 6 =(3p+ 1− 12− 12+p− 12(p+ 1))ε∫W p+1 +O(ε2−)= − p− 32(p+ 1)ε∫W p+1 +O(ε2−)which is negative for 3 < p < 5 and ε > 0 and small. We note that whenp = 3, this leading order term vanishes.Lemma 3.2.3. Take 3 < p < 5. Denote by V = Vε a non-negative, radially-symmetric minimizer for (3.30) with ω = ω(ε) (as established in Lemma693.2. Variational Characterization3.2.2). Then for any εj → 0, Vεj is a minimizing sequence for the (unper-turbed) variational problemS0,0(W ) = min{S0,0(u) | u ∈ H˙1 \ {0}, K0(u) = 0} (3.32)in the sense thatK0(Vεj )→ 0, lim supε→0S0,0(Vεj ) ≤ S0,0(W ).Proof. Since0 = Kε(V ) = K0(V )− 3(p− 1)2(p+ 1)ε∫V p+1,and by Lemma 3.2.2,S0,0(W ) > mε,ω(ε) = Sε,ω(ε)(V ) = S0,0(V )−1p+ 1ε∫V p+1 +12ω∫V 2,(3.33)the lemma will be implied by the claim:ε∫V p+1 → 0 as ε→ 0. (3.34)To address the claim, first introduce the functionalIε,ω(u) := Sε,ω(u)− 13Kε(u)=16∫|∇u|2 + 16∫|u|6 + p− 32(p+ 1)ε∫|u|p+1 + 12ω∫|u|2and observe that since Kε(V ) = 0,Iε,ω(ε)(V ) = Sε,ω(V ) < S0,0(W )and so the following quantities are all bounded uniformly in ε:∫|∇V |2,∫V 6, ε∫V p+1, ω∫V 2 . 1.By interpolationε∫V p+1 ≤ ε‖V ‖(5−p)/2L2‖V ‖3(p−1)/2L6. εω−(5−p)/4(ω∫V 2)(5−p)/4.703.2. Variational CharacterizationSo (3.34) holds, provided that ε4/(5−p)  ω. Since ω ∼ ε2, this indeed holdsfor 3 < p < 5.With the claim in hand we can finish the argument. The fact thatK0(V )→ 0 now follows from Kε(V ) = 0. Also, from Lemma 3.2.2 we knowthat for ε ≥ 0S0,0(V )− εp+ 1∫V p+1 ≤ Sε,ω(V ) ≤ S0,0(W )and so lim supε→0 S0,0(V ) ≤ S0,0(W ).Lemma 3.2.4. For a sequence εj ↓ 0, let V = Vεj be corresponding non-negative, radially-symmetric minimizers of (3.30) with ω = ω(εj). There isa subsequence εjk and a scaling µ = µk such that along the subsequence,V µ = µ1/2V (µ·)→ νWin H˙1 with ν = 1.Proof. The result with ν = 1 or ν = 0 follows from the bubble decompositionof Ge´rard [37] (see eg. the notes of Killip and Vis¸an [58], in particular The-orem 4.7 and the proof of Theorem 4.4). Therefore we need only eliminatethe possibility that ν = 0.If ν = 0 then∫ |∇Vε|2 → 0 (along the given subsequence). Then by theSobolev inequality,0 = Kε(Vε) = (1 + o(1))∫|∇Vε|2 − 3(p− 1)2(p+ 1)ε∫V p+1ε ,and so ∫|∇Vε|2 . ε∫V p+1ε .However, we have already seen∫|∇Vε|2 . ε∫V p+1ε . εω−(5−p)/4(ω∫V 2ε)(5−p)/4(∫|∇Vε|2)3(p−1)/4via interpolation and so(∫|∇Vε|2)(7−3p)/4. εω−(5−p)/4(ω∫V 2ε)(5−p)/4→ 0as above. Note that (7 − 3p)/4 = −3(p − 7/3)/4 < 0. Hence ν = 0 isimpossible and so we conclude that ν = 1. The result follows.713.2. Variational CharacterizationRemark 3.2.5. This lemma implies in particular that for V = Vε, ω = ω(ε),S0,0(V ) = S0,0(Vµ)→ S0,0(W ), and so by (3.33) and (3.34),ω∫V 2 → 0.Remark 3.2.6. Note that V µ is a minimizer of the minimization problem(3.30), and a solution to (3.31), with ε and ω replaced withε˜ = µ5−p2 ε, ω˜ = µ2ω.Under this scaling the following properties are preserved:ε˜45−p = µ2ε45−p  µ2ω = ω˜ε˜∫(V µ)p+1 = ε∫V p+1 → 0ω˜∫(V µ)2 = ω∫V 2 → 0.Moreover,ε˜→ 0, ω˜ → 0,the latter since otherwise ‖V µ‖L2 → 0 along some subsequence, contradictingV µ →W 6∈ L2 in H˙1, and then the former by the first relation above.Lemma 3.2.7. LetV µ = W + η˜, ‖η˜‖H˙1 → 0, ε˜→ 0be a sequence as provided by Lemma 3.2.4. There is a further scalingν = νε˜ = 1 + o(1)so that(V µ)ν = W ν + η˜ν =: W + ηˆretains ‖ηˆ‖H˙1 → 0, but also satisfies the orthogonality condition0 = 〈R0(−ωˆ)V ψ,F(εˆ, ωˆ, ηˆ)〉 (3.35)with the corresponding εˆ = ν(5−p)/2ε˜ and ωˆ = ν2ω˜.723.2. Variational CharacterizationProof. We may rewrite the above inner-product as〈R0(−ωˆ)V ψ,F(εˆ, ωˆ, ηˆ)〉 = −5√3〈ηˆ, (H + ωˆ)R0(−ωˆ)W 4ΛW 〉and observe from the resonance equation (1.18)5R0(−ωˆ)W 4ΛW = ΛW − ωˆR0(−ωˆ)ΛWand so(H + ωˆ)R0(−ωˆ)W 4ΛW = (−∆ + ωˆ − 5W 4)R0(−ωˆ)W 4ΛW= (1− 5W 4R0(−ωˆ))W 4ΛW = ωˆW 4R0(−ωˆ)ΛWso the desired orthogonality condition reads0 =1√ωˆ〈ηˆ, (H + ωˆ)R0(−ωˆ)W 4ΛW 〉 =√ωˆ〈W ν −W + η˜ν ,W 4R0(−ωˆ)ΛW 〉.Now since ΛW = ddµWµ|µ=1, by Taylor expansion‖W ν −W − (ν − 1)ΛW‖L6 . (ν − 1)2,and using (3.7)‖W 4R0(−ωˆ)ΛW‖L65. ‖R0(−ωˆ)ΛW‖L∞ . 1√ωˆ,we arrive at0 = (ν − 1)(√ωˆ〈ΛW, W 4R0(−ωˆ)ΛW 〉)+O((ν − 1)2) +O(‖η˜ν‖L6)Computations exactly as for (3.19) lead to√ωˆ〈ΛW, W 4R0(−ωˆ)ΛW 〉 = 6√3pi5+O(√ωˆ),and so the desired orthogonality condition reads0 = (ν − 1)(1 + o(1)) +O((ν − 1)2) +O(‖η˜ν‖L6)which can therefore be solved for ν = 1 + o(1) using ‖η˜ν‖L6 = o(1).733.2. Variational CharacterizationThe functionsWεˆ := (Vµ)ν = W + ηˆproduced by Lemma 3.2.7 solve the minimization problem (3.30), and thePDE (3.31), with ε and ω replaced (respectively) by εˆ → 0 and ωˆ. Sinceνε˜ = 1 + o(1), the propertiesεˆ45−p  ωˆ → 0, εˆ∫W p+1εˆ → 0, ωˆ∫W 2εˆ → 0persist.It remains to show that that Vεˆ agrees with Qεˆ constructed in Theo-rem 1.7.2. First:Lemma 3.2.8. For 3 < r ≤ ∞, and εˆ sufficiently small,‖ηˆ‖Lr . εˆ+√ωˆ1− 3r .Proof. Since Wεˆ is a solution of (1.11), the remainder ηˆ must satisfy (3.4).So‖ηˆ‖Lr = ‖(H + ωˆ)−1 (−ωˆW + εˆf(W ) +N(ηˆ)) ‖Lr. εˆ+√ωˆ1− 3r + ‖R0(−ωˆ)N(ηˆ)‖Lrusing (3.35) and after observing the computations of Lemma 3.1.9. We nowestablish the required bounds on the remainder, beginning with 3 < r <∞.Let q ∈ (1, 32) satisfy 1q − 1r = 23 :• ‖R0(−ωˆ)εˆW p−1ηˆ‖Lr . εˆ‖W p−1ηˆ‖Lq . εˆ‖W‖p−1L32 (p−1)‖ηˆ‖Lr . εˆ‖ηˆ‖Lr• ‖R0(−ωˆ)εˆηˆp‖Lr . εˆωˆ−5−p4 ‖ηˆp‖L6r6+(p−1)r. o(1)‖ηˆ‖p−1L6‖ηˆ‖Lr. o(1)‖ηˆ‖Lr• ‖R0(−ωˆ)W 3ηˆ5‖Lr . ‖W 3ηˆ2‖Lq . ‖W‖3L6‖ηˆ‖L6‖ηˆ‖Lr . o(1)‖ηˆ‖Lr• ‖R0(−ωˆ)ηˆ5‖Lr . ‖ηˆ5‖Lq . ‖ηˆ‖4L6‖ηˆ‖Lr . o(1)‖ηˆ‖Lrwhere in the second inequality we used ωˆ  εˆ4/(5−p). Combining, we haveachieved‖ηˆ‖Lr . εˆ+√ωˆ1− 3r + o(1)‖ηˆ‖Lrand so obtain the desired estimate for 3 < r < ∞. It remains to deal withr = ∞. The first three estimates proceed similarly, while the last one usesthe now-established Lr estimate:743.2. Variational Characterization• ‖R0(−ωˆ)εˆW p−1ηˆ‖L∞ . εˆ‖W p−1ηˆ‖L32−∩L 32+. εˆ‖W‖p−1L32 (p−1)−∩L 32 (p−1)+‖ηˆ‖L∞ . εˆ‖ηˆ‖L∞• ‖R0(−ωˆ)εˆηˆp‖L∞ . εˆωˆ−5−p4 ‖ηˆp‖L6p−1. o(1)‖ηˆ‖p−1L6‖ηˆ‖L∞. o(1)‖ηˆ‖L∞• ‖R0(−ωˆ)W 3ηˆ5‖L∞ . ‖W 3ηˆ2‖L32−∩L 32+ . ‖W‖3L6−∩L6+‖ηˆ‖L6‖ηˆ‖L∞. o(1)‖ηˆ‖L∞• ‖R0(−ωˆ)ηˆ5‖L∞ . ‖ηˆ5‖L32−∩L 32+ . ‖ηˆ‖4L6−∩L6+‖ηˆ‖L∞. (εˆ+ ωˆ 14−)4‖ηˆ‖L∞ . o(1)‖ηˆ‖L∞which, combined, establish the desired estimate with r =∞. Strictly speak-ing, these are a priori estimates, since we do not know ηˆ ∈ Lr for r > 6 tobegin with. However, the typical argument of performing the estimates ona series of smooth functions that approximate η remedies this after passingto the limit.Lemma 3.2.9. Write ωˆ = λˆ2. For εˆ sufficiently small, ‖ηˆ‖L∞ . εˆ, andλˆ = λ(εˆ, ηˆ) as given in Lemma 3.1.6. Moreover, Wεˆ = W + ηˆ = Qεˆ.Proof. From the orthogonality equation (3.35),0 = 〈R0(−λˆ2)V ψ,−λˆ2W + εˆW p +N(ηˆ)〉= −λˆ · λˆ〈R0(−λˆ2)V ψ,W 〉+ εˆ〈R0(−λˆ2)V ψ,W p〉+ 〈R0(−λˆ2)V ψ,N(ηˆ)〉.Now re-using estimates (3.18) and (3.19), as well as• |〈R0(−λˆ2)V ψ,W 3ηˆ2〉| . ‖R0(−λˆ2)V ψ‖L6‖W 3ηˆ2‖L6/5. ‖V ψ‖L6/5‖ηˆ‖2L∞‖W 3‖L6/5 . ‖ηˆ‖2L∞• |〈R0(−λˆ2)V ψ, ηˆ5〉| . ‖R0(−λˆ2)V ψ‖L6‖ηˆ5‖L 65. ‖V ψ‖L65‖ηˆ‖5L6 . ‖ηˆ‖5L6• |〈R0(−λˆ2)V ψ, εˆηˆp〉| . εˆ‖R0(−λˆ2)V ψ‖L6‖ηˆp‖L 65. εˆ‖V ψ‖L65‖ηˆ‖pL65 p. εˆ · ‖ηˆ‖pL65 p• |〈R0(−λˆ2)V ψ, εˆW p−1ηˆ〉| . εˆ‖R0(−λˆ2)V ψ‖L6‖W p−1ηˆ‖L 65. εˆ‖V ψ‖L65‖W p−1‖L32‖ηˆ‖L6 . εˆ‖ηˆ‖L6 ,753.2. Variational Characterizationcombined with Lemma 3.2.8, yields(λˆ− λ(1)εˆ)(1 +O(λˆ1−)) = O(λˆ2 + εˆ2 + εˆλˆ 12 )from which followsλˆ− λ(1)εˆ εˆ,and then by Lemma 3.2.8 again,‖ηˆ‖Lr . εˆ1− 3r , 3 < r ≤ ∞.It now follows from Lemma 3.1.6 that λˆ = λ(εˆ, ηˆ) for εˆ small enough.Finally, the uniqueness of the fixed-point in the L∞-ball of radius Rεˆfrom Lemma 3.1.9 implies that Wεˆ = Qεˆ, where Qεˆ is the solution con-structed in Theorem 1.7.2.We have so far established that, up to subsequence, and rescaling, asequence of minimizers Vεj eventually coincides with a solution Qε as con-structed in Theorem 1.7.2: ξ1/2j Vεj (ξj ·) = Qεˆj (here ξj = νjµj). It remainsto remove the scaling ξj and establish that εˆj = εj :Lemma 3.2.10. Suppose V (x) = ξ−12Qεˆ(x/ξ) solves (3.31) with ω = ω(ε)(as given in Theorem 1.7.2), where εˆ = ξ(5−p)/2ε, ωˆ = ξ2ω, and ωˆ = ω(εˆ).Then ξ = 1 and εˆ = ε, and so V = Qε.Proof. By assumption ωˆ = ξ2ω(ε) = ω(εˆ), soω(ε) = Ωε(εˆ), Ωε(εˆ) :=(εεˆ)4/(5−p)ω(εˆ).This relation is satisfied if εˆ = ε (ξ = 1), and our goal is to show it is notsatisfied for any other value of εˆ. Thus we will be done if we can showthat Ωε is monotone in εˆ. Take ε1 and ε2 with 0 < ε1 < ε2 ≤ ε0. Letα = 4/(5− p) > 2 and assume that 0 < ε2 − ε1  ε1. Denoting ω(εj) = λ2j ,we estimate:ε−α (Ωε(ε2)− Ωε(ε1)) = ε−α2 λ22 − ε−α1 λ21= ε−α2 (λ2 − λ1)(λ2 + λ1) + λ21(ε−α2 − ε−α1)≈ ε−α1 λ(1)(ε2 − ε1) · 2λ(1)ε1+ ε21(λ(1))2ε−α1(− αε1(ε2 − ε1) +O((ε2 − ε1ε1)2))≈ ε1−α1 (λ(1))2(ε2 − ε1) (2− α)< 0763.2. Variational Characterizationwhere we have used Lemma 3.27. With the monotonicity argument completewe conclude that ε = εˆ and ξ = 1 so there follows V = Qε.The remaining lemma completes the proof of Theorem 1.7.7:Lemma 3.2.11. There is ε0 > 0 such that for 0 < ε ≤ ε0 and ω = ω(ε), thesolution Qε of (3.31) constructed in Theorem 1.7.2 is the unique positive,radially symmetric solution of the minimization problem (3.30).Proof. This is the culmination of the previous series of Lemmas. We knowthat minimizers V = Vε exist by Lemma 3.2.2. Arguing by contradiction, ifthe statement is false, there is a sequence Vεj , εj → 0, of such minimizers,for which Vεj 6= Qεj . We apply Lemmas 3.2.3, 3.2.4, 3.2.7, 3.2.8 and 3.2.10in succession to this sequence, to conclude that along a subsequence, Vεjand Qεj eventually agree, a contradiction.Finally, for a given ε, we establish a range of ω for which a minimizerexists and is, up to scaling, a constructed solution. This addresses Remark1.7.9.Corollary 3.2.12. Fix ε > 0 and take ω ∈ [ω,∞) whereω = ε4/(5−p)ε−4/(5−p)0 ω(ε0) ≤ ω(ε).The minimization problem (3.30) with ε and ω has a solution Q given byQ(x) = µ1/2Qεˆ(µx)where Qεˆ is a constructed solution with 0 < εˆ ≤ ε0 and corresponding ω(εˆ).The scaling factor, µ, satisfies the relationshipsε = εˆµ(5−p)/2, ω = ω(εˆ)µ2.Proof. Fix ε > 0. Take any 0 < εˆ ≤ ε0 and corresponding constructed ω(εˆ)and constructed solution Qεˆ. Then, for scaling µ = (ε/εˆ)2/(5−p) the functionQ(x) = µ1/2Qεˆ(µx)is a solution to the elliptic problem (3.31) with ε and ω = ω(εˆ)µ2. Recallfrom Lemma 3.2.10 that ω(εˆ)µ2 is monotone in εˆ. Taking εˆ ↓ 0 yieldsω →∞. Setting εˆ = ε0 yields ω = ω.In other words if we fix ε and ω ∈ [ω,∞) from the start we determinean εˆ and µ that generate the desired Q. We claim that the function Q(x)773.3. Dynamics Below the Ground Statesis a minimizer of the problem (3.30) with ε and ω. Suppose not. Thatis, suppose there exists a function 0 6= v ∈ H1 with Kε(v) = 0 such thatSε,ω(v) < Sε,ω(Q). Set w(x) = µ−1/2v(µ−1x) and note that 0 = Kε(v) =Kεˆ(w). We now seeSεˆ,ω(εˆ)(w) = Sε,ω(v) < Sε,ω(Q) = Sεˆ,ω(εˆ)(Qεˆ)which contradicts the fact that Qεˆ is a minimizer of the problem (3.30) withεˆ and ω(εˆ). Therefore, Q(x) is a minimizer of (3.30) with ε and ω, whichconcludes the proof.3.3 Dynamics Below the Ground StatesIn this final section we establish Theorem 1.7.10, the scattering/blow-updichotomy for the perturbed critical NLS (2.1).We begin by summarizing the local existence theory for (2.1). This isbased on the classical Strichartz estimates for the solutions of the homoge-neous linear Schro¨dinger equationi∂tu = −∆u, u|t=0 = φ ∈ L2(R3) =⇒ u(x, t) = eit∆φ ∈ C(R, L2(R3))and the inhomogeneous linear Schro¨dinger equation (with zero initial data)i∂tu = −∆u+ f(x, t), u|t=0 = 0 =⇒ u(x, t) = −i∫ t0ei(t−s)∆f(·, s)ds :‖eit∆φ‖S(R) ≤ C‖φ‖L2(R3),∥∥∥∥∫ t0ei(t−s)∆f(·, s)ds∥∥∥∥S(I)≤ C‖f‖N(I),(3.36)where we have introduced certain Lebesgue norms for space-time functionsf(x, t) on a time interval t ∈ I ⊂ R:‖f‖LrtLqx(I) =∥∥‖f(·, t)‖Lq(R3)∥∥Lr(I) ,‖f‖S(I) := ‖f‖L∞t L2x(I)∩L2tL6x(I), ‖f‖N(I) := ‖f‖L1tL2x(I)+L2tL65x (I),together with the integral (Duhamel) reformulation of the Cauchy prob-lem (2.1):u(x, t) = eit∆u0 + i∫ t0ei(t−s)∆(|u|4u+ ε|u|p−1u)ds783.3. Dynamics Below the Ground Stateswhich in particular gives the sense in which we consider u(x, t) to be asolution of (2.1). This lemma summarizing the local theory is standard(see, for example [15, 56]):Lemma 3.3.1. Let 3 ≤ p < 5, ε > 0. Given u0 ∈ H1(R3), there is a uniquesolution u ∈ C((−Tmin, Tmax);H1(R3)) of (2.1) on a maximal time intervalImax = (−Tmin, Tmax) 3 0. Moreover:1. space-time norms: u,∇u ∈ S(I) for each compact time interval I ⊂Imax;2. blow-up criterion: if Tmax < ∞, then ‖u‖L10t L10x ([0,Tmax)) = ∞ (withsimilar statement for Tmin);3. scattering: if Tmax = ∞ and ‖u‖L10t L10x ([0,∞)) < ∞, then u scatters(forward in time) to 0 in H1:∃ φ+ ∈ H1(R3) s.t. ‖u(·, t)− eit∆φ+‖H1 → 0 as t→∞(with similar statement for Tmin);4. small data scattering: for ‖u0‖H1 sufficiently small, Imax = R, ‖u‖L10t L10x (R) .‖∇u0‖L2, and u scatters (in both time directions).Remark 3.3.2. The appearance here of the L10t L10x space-time norm is nat-ural in light of the Strichartz estimates (3.36). Indeed, interpolation betweenL∞t L2x and L2tL6 shows that‖eit∆φ‖LrtLqx(R) . ‖φ‖L2 ,2r+3q=32, 2 ≤ r ≤ ∞(such an exponent pair (r, q) is called admissible), so then if ∇φ ∈ L2, by aSobolev inequality,‖eit∆φ‖L10x . ‖∇eit∆φ‖L 3013x ∈ L10t ,since (r = 10, q = 3013) is admissible.The next lemma is a standard extension of the local theory called aperturbation or stability result, which shows that any ‘approximate solution’has an actual solution remaining close to it. In our setting (see [56, 78]):793.3. Dynamics Below the Ground StatesLemma 3.3.3. Let u˜ : R3 × I → C be defined on time interval 0 ∈ I ⊂ Rwith‖u˜‖L∞t H1x(I)∩L10t L10x (I) ≤M,and suppose u0 ∈ H1(R3) satisfies ‖u0‖L2 ≤M . There exists δ0 = δ0(M) >0 such that if for any 0 < δ < δ0, u˜ is an approximate solution of (2.1) inthe sense‖∇e‖L107t L107x (I)≤ δ, e := i∂tu˜+ ∆u˜+ |u˜|4u˜+ ε|u˜|p−1u˜,with initial data close to u0 in the sense‖∇ (u˜(·, 0)− u0) ‖L2 ≤ δ,then the solution u of (2.1) with initial data u0 has Imax ⊃ I, and‖∇ (u− u˜) ‖S(I) ≤ C(M)δ.Remark 3.3.4. The space-time norm ∇e ∈ L107t L107x in which the error ismeasured is natural in light of the Strichartz estimates (3.36), since L107t L107xis the dual space of L103t L103x , and (103 ,103 ) is an admissible exponent pair.Given a local existence theory as above, an obvious next problem is to de-termine if the solutions from particular initial data u0 are global (Imax = R),or exhibit finite-time blow-up (Tmax < ∞ and/or Tmin < ∞). Theo-rem 1.7.10 solves this problem for radially-symmetric initial data lying ‘be-low the ground state’ level of the action: for any ε > 0, ω > 0, setmε,ω := inf{Sε,ω(u) | u ∈ H1(R3) \ {0},Kε(u) = 0} (3.37)(see (1.22) for expressions for the functionals Sε,ω and Kε), and note thatfor ε 1 and ω = ω(ε), by Theorem 1.7.7 we have mε,ω = Sε,ω(Qε). Fromhere on, we fix a choice ofε > 0, ω > 0, p ∈ (3, 5)(though some results discussed below extend to p ∈ (73 , 5)):Theorem 3.3.5. Let u0 ∈ H1(R3) be radially-symmetric and satisfySε,ω(u0) < mε,ω,and let u be the corresponding solution to (2.1):803.3. Dynamics Below the Ground States1. If Kε(u0) ≥ 0, u is global, and scatters to 0 as t→ ±∞;2. if Kε(u0) < 0, u blows-up in finite time (in both time directions).Remark 3.3.6. The argument which gives the finite-time blow-up (the sec-ond statement) is classical, going back to [61, 62]. It rests on the followingingredients: conservation of mass and energy imply Sε,ω(u) ≡ Sε,ω(u0) <mε,ω, so that the condition Kε(u) < 0 is preserved (by definition of mε,ω); aspatially cut-off version of the formal variance identity for (NLS)d2dt212∫|x|2|u(x, t)|2dx = ddt∫x · = (u¯∇u) dx = 2Kε(u) ; (3.38)and exploitation of radial symmetry to control the errors introduced by thecut-off. In fact, a complete argument in exactly our setting is given as theproof of Theorem 1.3 in [1] (it is stated there for dimensions ≥ 4 but in factthe proof covers dimension 3 as well). So we will focus here only on theproof of the first (scattering) statement.The concentration-compactness approach of Kenig-Merle [54] to provingthe scattering statement is by now standard. In particular, [2] provides acomplete proof for the analogous problem in dimensions ≥ 5. In fact, theproof there is more complicated for two reasons: there is no radial symmetryrestriction; and in dimension n, the corresponding nonlinearity includes theterm |u|p−1u with p > 1 + 4n loses smoothness, creating extra technical dif-ficulties. We will therefore provide just a sketch of the (simpler) argumentfor our case, closely following [56], where this approach is implemented forthe defocusing quintic NLS perturbed by a cubic term, and taking the ad-ditional variational arguments we need here from [1, 2], highlighting pointswhere modifications are needed.In the next lemma we recall some standard variational estimates forfunctions with action below the ground state level mε,ω. The idea goes backto [63], but proofs in this setting are found in [1, 2]. Recall the ‘unperturbed’ground state level is attained by the Aubin-Talenti function W :m0,0 := E0(W ) = inf{E0(u) | u ∈ H1(R3) \ {0},K0(u) = 0},and introduce the auxilliary functionalIω(u) := Sε,ω(u)− 23(p− 1)Kε(u)=p− 732(p− 1)∫|∇u|2 + 5− p6(p− 1)∫|u|6 + 12ω∫|u|2813.3. Dynamics Below the Ground Stateswhich is useful since all its terms are positive, and noteKε(u) ≥ 0 =⇒ ‖u‖2H1 . Iω(u) ≤ Sε,ω(u). (3.39)Define, for 0 < m∗ < mε,ω, the setAm∗ := {u ∈ H1(R3) | Sε,ω(u) ≤ m∗, Kε(u) > 0}and note that it is is preserved by (2.1):u0 ∈ Am∗ =⇒ u(·, t) ∈ Am∗ for all t ∈ Imax.Indeed, by conservation of mass and energy Sε,ω(u(·, t)) = Sε,ω(u0) ≤ m∗.Moreover if for some t0 ∈ Imax, Kε(u(·, t0)) ≤ 0, then by H1 continuityof u(·, t) and of Kε, we must have Kε(u(·, t1)) = 0 for some t1 ∈ Imax,contradicting m∗ < mε,ω.Lemma 3.3.7. 1. mε,ω ≤ m0,0, and (3.37) admits a minimizer if mε,ω <m0,0;2. we havemε,ω = inf{Iω(u) | u ∈ H1(R3) \ {0},Kε(u) ≤ 0}, (3.40)and a minimizer for this problem is a minimizer for (3.37), and viceversa;3. given 0 < m∗ < mε,ω, there is κ(m∗) > 0 such thatu ∈ Am∗ =⇒ Kε(u) ≥ κ(m∗) > 0. (3.41)After the local theory, and in particular the perturbation Lemma 3.3.3,the key analytical ingredient is a profile decomposition, introduced into theanalysis of critical nonlinear dispersive PDE by [5, 55]. This version, takenfrom [56] (and simplified to the radially-symmetric setting), can be thoughtof as making precise the lack of compactness in the Strichartz estimates forH˙1(R3) data, when the data is bounded in H1(R3):Lemma 3.3.8. ([56], Theorem 7.5) Let {fn}∞n=1 be a sequence of radiallysymmetric functions, bounded in H1(R3). Possibly passing to a subsequence,there is J∗ ∈ {0, 1, 2, . . .} ∪ {∞} such that for each finite 1 ≤ j ≤ J∗ thereexist (radially symmetric) ‘profiles’ φj ∈ H˙1 \ {0}, ‘scales’ {λjn}∞n=1 ⊂ (0, 1],and ‘times’ {tjn}∞n=1 ⊂ R satisfying, as n→∞,λjn ≡ 1 or λjn → 0, tjn ≡ 0 or tjn → ±∞.823.3. Dynamics Below the Ground StatesIf λjn ≡ 1 then additionally φj ∈ L2(R3). For some 0 < θ < 1, defineφjn(x) :=[eitjn∆φj](x) λjn ≡ 1(λjn)− 12[eitjn∆P≥(λjn)θφj] (xλjn)λjn → 0,where P≥N denotes a standard smooth Fourier multiplier operator (Littlewood-Paley projector) which removes the Fourier frequencies ≤ N . Then for eachfinite 1 ≤ J ≤ J∗ we have the decompositionfn =J∑j=1φjn + wJnwith:• small remainder: limJ→J∗lim supn→∞‖eit∆wJn‖L10t L10x (R) = 0• decoupling: for each J , limn→∞[M(fn)−J∑j=1M(φjn)−M(wJn)]= 0,and the same statement for the functionals Eε, Kε, Sω,ε and Iω;• orthogonality: limn→∞[λjnλkn+ λknλjn+ |tjn(λjn)2−tkn(λkn)2|λjnλkn]=∞ for j 6= k.The global existence and scattering statement 1 of Theorem 1.7.10 isestablished by a contradiction argument. For 0 < m < mε,ω, setτ(m) := sup{‖u‖L10t L10x (Imax) | Sε,ω(u0) ≤ m, Kε(u0) > 0}where the supremum is taken over all radially-symmetric solutions of (2.1)whose data u0 satisfies the given conditions. It follows from the local theoryabove that τ is non-decreasing, continuous function of m into [0,∞], andthat τ(m) < ∞ for sufficiently small m (by part 4 of Lemma 3.3.1). Byparts 2-3 of Lemma 3.3.1, if τ(m) <∞ for all m < mε,ω, the first statementof Theorem 1.7.10 follows. So we suppose this is not the case, and that infactm∗ := sup{m | 0 < m < mε,ω, τ(m) <∞} < mε,ω.By continuity, τ(m∗) =∞, and so there exists a sequence un(x, t) of global,radially-symmetric solutions of (2.1) satisfyingSε,ω(un) ≤ m∗, Kε(un(·, 0)) > 0, (3.42)833.3. Dynamics Below the Ground Statesandlimn→∞ ‖un‖L10t L10x ([0,∞)) = limn→∞ ‖un‖L10t L10x ((−∞,0]) =∞ (3.43)(the last condition can be arranged by time shifting, if needed). The ideais to pass to a limit in this sequence in order to obtain a solution sitting atthe threshold action m∗.Lemma 3.3.9. There is a subsequence (still labelled un) such that un(x, 0)converges in H1(R3).Proof. This is essentially Proposition 9.1 of [56], with slight modificationsto incorporate the variational structure. We give a brief sketch. The se-quence un(·, 0) is bounded in H1 by (3.39), so we may apply the profiledecomposition Lemma 3.3.8: up to subsequence,un(·, 0) =J∑j=1φjn + wJn .If we can show there is only one profile (J∗ = 1), that λ1n ≡ 1, t1n ≡ 0,and that w1n → 0 in H1, we have proved the lemma. By (3.42) and thedecoupling,m∗ − 23(p− 1)κ(m∗) ≥ Sε,ω(un(·, 0))− 23(p− 1)Kε(un(·, 0))= Iω(un(·, 0)) =J∑j=1Iω(φjn) + Iω(wJn) + o(1),and since Iω is non-negative, we have, for n large enough, Iω(φjn) < m∗for each j and Iω(wJn) < m∗. Since m∗ < mε,ω, it follows from (3.40) thatKε(φjn) > 0 and Kε(wJn) ≥ 0, so also Sε,ω(φjn) > 0 and Sε,ω(wJn) ≥ 0. Henceif there is more than one profile, by the decouplingm∗ ≥ Sε,ω(un(·, 0)) =J∑j=1Sε,ω(φjn) + Sε,ω(wJn) + o(1),we have, for each j, and n large enough, for some η > 0,Sε,ω(φjn) ≤ m∗ − η, Kε(φjn) > 0. (3.44)Following [56], we introduce nonlinear profiles vjn associated to each φjn.843.3. Dynamics Below the Ground StatesFirst, suppose λjn ≡ 1. If tjn ≡ 0, then vjn = vj is defined to be the solutionto (2.1) with initial data φj . If tjn → ±∞, vj is defined to be the solutionscattering (in H1) to eit∆φj as t→ ±∞, and vjn(x, t) := vj(t+ tjn). In bothcases, it follows from (3.44) that vjn is a global solution, with ‖vjn‖L10t L10x (R) ≤τ(m∗ − η) <∞.For the case λjn → 0, we simply let vjn be the solution of (2.1) with initialdata φjn. As in [56] Proposition 8.3, vjn is approximated by the solution u˜jnof the unperturbed critical NLS (1.13) (since the profile is concentrating,the sub-critical perturbation ‘scales away’) with data φjn (or by a scatteringprocedure in case tnj → ±∞). The key additional point here is that by (3.44),and since m∗ < mε,ω ≤ m0,0, it follows that for n large enoughE0(vjn) ≤ m∗ < m0,0, K0(vjn) > 0,and so by [54], u˜jn is a global solution of (1.13), with ‖u˜jn‖L10t L10x (R) ≤C(m∗) < ∞. It then follows from Lemma 3.3.3 that the same is true ofvjn.These nonlinear profiles are used to construct what are shown in [56] tobe increasingly accurate (for sufficiently large J and n) approximate solu-tions in the sense of Lemma 3.3.3,uJn(x, t) :=J∑j=1vjn(x, t) + eit∆wJnwhich are moreover global with uniform space-time bounds. This contra-dicts (3.43).Hence there is only one profile: J∗ = 1, and the decoupling also implies‖w1n‖H1 → 0. Finally, the possibilities t1n → ±∞ or λ1n → 0 are excludedjust as in [56], completing the argument.Given this lemma, let u0 ∈ H1(R3) be the H1 limit of (a subsequence) ofun(x, 0), and let u(x, t) be the corresponding solution of (2.1) on its maximalexistence interval Imax 3 0. We see Sε,ω(u) = Sε,ω(u0) ≤ m∗. Whether u isglobal or not, it follows from Lemma 3.3.1 (part 2), (3.43) and Lemma 3.3.3,that‖u‖L10t L10x (Imax) =∞, hence also Sε,ω(u) = m∗.It follows also that{u(·, t) | t ∈ Imax} is a pre-compact set in H1(R3).853.3. Dynamics Below the Ground StatesTo see this, let {tn}∞n=1 ⊂ Imax, and note that since‖u‖L10t L10x ((−Tmin,tn]) = ‖u‖L10t L10x ([tn,Tmax)) =∞,and so (the proof of) Lemma 3.3.9 applied to the sequence u(x, t−tn) impliesthat {u(x, tn)} has a convergent subsequence in H1.The final step is to show that this ‘would-be’ solution u with these specialproperties, sometimes called a critical element cannot exist. For this, firstnote that u must be global: Imax = R. This is because if, say, Tmax < ∞,then for any tn → Tmax−, u(·, tn) → u˜0 ∈ H1(R3) (up to subsequence)in H1, by the pre-compactness. Then by comparing u with the solution u˜of (2.1) with initial data u˜0 at t = tn using Lemma (3.3.3), we conclude thatu exists for times beyond Tmax, a contradiction.Finally, the possible existence of (the now global) solution u is ruled outvia a suitably cut-off version of the virial identity (3.38), using (3.41), andthe compactness to control the errors introduced by the cut-off, exactly asin [56] (Proposition 10, and what follows it). 86Chapter 4Directions for Future StudyWe collect here some suggestions (and speculations) for future problemsrelating to the results of Chapters 2 and 3.4.1 Improvements and Extensions of the CurrentWorkAs mentioned in Section 1.6 the asymptotic stability for the 1D mass sub-critical (p 6= 3) NLS is still open. This problem presents serious challenges.A successful attempt on this problem may involve detailed knowledge of thespectrum of the linearized operator as obtained in Chapter 2.The results of Chapter 3 apply only to dimension 3. The works [1,2] addressed the dynamics (scattering/blow-up) for the perturbed energycritical NLS (below the threshold) in dimensions n ≥ 4. Nevertheless itwould be interesting to see if the construction of Section 3.1 can be achievedin 4D. One would need to redo the resolvent estimates in Section 3.1.2 using[46] in place of [47], as well as perform many of the computations in 4D sincewe have used the particular 3D free resolvent expansion (3.5) and particular3D Young’s inequality (3.6), (3.7) and so on. If the same methods produceconstructed solutions in 4D it’s possible we could complete the analysis ofSections 3.2 and 3.3 to achieve a dynamical theorem. While this theoremalready exists [1, 2] some comparison may be interesting.In dimensions n ≥ 5 the linearized operator (1.17) no longer has a reso-nance (as a resonance appearing in dimensions 5D and higher is impossible[45]) but does have an edge-eigenvalue. Replacing the role of [47] with [45]we may be able to use our methods to achieve the results of Chapter 3 in5D and higher.One important case we are missing from Sections 3.2 and 3.3 is the casewhen p = 3. Of course, the cubic-quintic NLS is important for applicationsand seems as well to present interesting mathematical difficulties. Whilewe are able to construct solitary wave solutions in Section 3.1 we cannotdemonstrate these solutions to be ground states in Section 3.2. In Lemma874.2. Small Solutions to the Gross-Pitaevskii Equation3.2.2 the leading order in the computation of Sε,ω(Q) is zero when p = 3 andthe next to leading order is difficult to resolve. If we were able to at leastdemonstrate the existence of a ground state we could run the arguments ofSection 3.3 to achieve our dynamical theorem for p = 3. (although a groundstate in this case may simply not exist). Supposing that a ground statesolution does exist for p = 3 we would still have difficulty to demonstrateour constructed solutions as the ground state (it also simply may not be) asthe series of Lemmas 3.2.3, 3.2.4, 3.2.7, 3.2.8 and 3.2.10 in Section 3.2 allrequire 3 < p < 5. One could look for additional boundary case argumentsto add or else try to demonstrate that our constructed solution is not aground state.4.2 Small Solutions to the Gross-PitaevskiiEquationA distinct but related problem is to consider solutions to the Gross-Pitaevskiiequation (the nonlinear Schro¨dinger equation with a potential)i∂tψ = (−∆ + V )ψ ± |ψ|p−1ψ (4.1)with small initial data ψ(x, 0) = ψ0 ∈ H1 in n dimensions. Function V (x)is a potential which we can think of as a Schwartz (fast decaying) function.The power p should be taken mass super-critical but energy sub-critical toensure solutions do not blow-up in finite time. Global well-posedness is aresult of conservation of mass and energy and the smallness of initial data(see [15] Chapter 6). We think mainly about H1 results but there are alsoresults where further localizing assumptions (ψ0 ∈ L1) on the initial dataare made such as [66] [69] [70] [87].The simplest result is when −∆+V has no bound states nor a resonance.In this case all solutions of (4.1) will scatter; a result due to [49] and [73].The result with a single simple bound state was obtained for a class ofnonlinearities in [42] in 3D. This result was extended to 1D with supercriticalpower nonlinearity in [59]. In this situation the bound state of −∆ + Vgenerates nonlinear bound states of (4.1). These nonlinear bound statesgive rise to stable solitary waves. Both [42] and [59] establish that any smallsolution can be decomposed into a piece approaching a nonlinear boundstate and a piece that scatters. The state of the art is [27] where they treatthe 3D cubic defocusing equation with many simple eigenvalues. There aresome conditions on the relative values of the eigenvalues but the treatablecases are quite generic. While the result obtained is analogous to [42] and884.2. Small Solutions to the Gross-Pitaevskii Equation[59] the methods used in this paper differ from [42] and [59]. The paper[27] uses instead variational methods. By means of a Birkhoff normal formsargument they find an effective Hamiltonian which gives rise to a nonlinearFermi Golden Rule. The connection between the Hamiltonian structure anda Fermi Golden Rule was originally introduced in [24]. See also [25] for theintuition behind the argument. For us the special case of two eigenvalues in[27] will be most relevant. There are also some other results for two boundstates [71] [79] [80] [81] [82] which impose stronger restrictions on the initialdata and placement of the eigenvalues.Our questions revolve around the case when operator −∆ + V has aresonance. That is when (−∆ + V )φ1 = 0 for some φ1 /∈ L2 but φ1 ∈ Lq forsome 2 < q ≤ ∞ where q depends on the dimension n. For the statement ofquestions bellow we also assume that −∆+V also has a simple bound state,that is (−∆ + V )φ0 = e0φ0 with e0 < 0. We could, however, ask analogousquestions if −∆ +V has no bound state and just the resonance or if insteadof a resonance we have a simple threshold eigenvalue.Firstly, we may suspect that φ1 will generate nonlinear bound states.When H := −∆ + V has a bound sate the nonlinear problem (4.1) admitsa family of nonlinear bound states Q0[z] parametrized by z = (Q0, φ0) witheigenvalue E0 = E0[|z|]. This follows from standard bifurcation theory (seethe Appendix of [42]). The existence of nonlinear bound states comingfrom the resonance eigenfunction does not follow immediately from the trueeigenvalue result. If for example we writeQ1(x) = zφ1(x) + q(x) (4.2)then to have Q1 ∈ L2 we cannot have q ∈ L2. The Birman-Schwingertrick used in Chapter 2 does not work due to the presence of the nonlinearterm in (4.1) but we can instead proceed as in Chapter 3. A formal andpreliminary computation suggests that generically in the defocusing case theresonance will not yield nonlinear bound states while in the focusing case wewill have nonlinear bound states with corresponding eigenvalues close to thethreshold. The idea is to follow the analysis of Section 3.1 and substitute(4.2) to the nonlinear eigenvalue equation(H + λ2)Q1 ± |Q1|p−1Q1 = 0and write the resulting equation as a fixed point problem for qq = (H + λ2)−1f.894.2. Small Solutions to the Gross-Pitaevskii EquationHere f = f(λ, z, q). From [47] and [48] we have the resolvent expansion(H + λ2)−1 =1λ〈R0(λ)V φ1, ·〉φ1where R0(λ) = (−∆ + λ2)−1 is the free resolvent. The idea now is to solve0 = 〈R0(λ)V φ1, f〉q = (H + λ2)−1ffor λ = λ(z) via fixed point arguments where q is in a higher Lp space.Adjustments of the fixed point arguments of Section 3.1 may prove fruitfulin this setting.Secondly, we may consider the asymptotic stability of the ground statefamily e−iE0tQ0. One approach would be to proceed as in [42] but withweaker decay estimates. Taking the power in the nonlinearity higher mayhelp in this direction. An alternative, and perhaps more favourable, ap-proach to this problem would be to understand the spectrum of the linearizedoperator around Q0 in the presence of the resonance φ1. We comment onthis direction. Since we are interested in the stability of the family of groundstate solitary waves we consider a solution to (4.1) of the formψ(x, t) = e−iE0t(Q0(x) + ξ(x, t))where ξ is a small perturbation of Q0. Since Q0 is the ground state wecan take it positive and real. Substituting the above to (4.1) and removingknown information about Q0 yieldsi∂tξ = (H − E0)ξ ± (p− 1)Qp−10 Reξ +Qp−10 ξ +N(ξ)where N(ξ) is nonlinear in ξ. The above with N removed is the linearizedequation. We complexify by letting~ξ :=(ξξ¯)to seei∂t~ξ = L~ξwhereL :=(H − E0 ± p+12 Qp−10 ±p+12 Qp−10∓p+12 Qp−10 −(H − E0 ± p+12 Qp−10)) .904.2. Small Solutions to the Gross-Pitaevskii EquationThe unperturbed L0 for z = 0 then has a resonance at the threshold. IndeedL0 =(H − e0 00 − (H − e0))has essential spectrum (−∞, e0] ∪ [−e0,∞), a double eigenvalue at 0 and aresonance on each threshold. The question is then about the spectrum of L.The full operator in question has essential spectrum (−∞, E0]∪ [−E0,∞) aswell as the double eigenvalue at 0 but the fate of the resonance has yet to beseen. Again, a formal preliminary computation suggests that generically inthe focusing case we have a true eigenvalue close to the threshold and thatin the defocusing case the resonance disappears. In 3D we may be able toapply [28] which treats general perturbations of the linearized operator. In1D the experience gained from Chapter 2 may be of use. Once the spectrumof the operator L is understood we may be able to proceed as in [79–82] toobtain asymptotic stability results for the ground state as well as the excitedstate, in any case where it may exist. A difficulty in this series of paperswas the eigenvalues of H being close together or, after complexification, Lhaving eigenvalues close to zero. For us any non-zero eigenvalues of L willbe close to the threshold and so far from zero.Finally, we may wish to understand the long time behaviour of all smallsolutions of (4.1). That is obtain an asymptotic stability and completenesstheorem in the spirit of [42], [59], [27]. This is the most substantial andchallenging question present. The goal is to prove a theorem resembling thefollowing. For any small solution ψ of (4.1) we have the unique decomposi-tionψ(t) = Q0[z0(t)] +Q1[z1(t)] + η(t)in the presence of nonlinear bound states connected to the resonance orψ(t) = Q0[z0(t)] + η(t)in the absence of nonlinear bound states connected to the resonance. In theabove zj and η must also enjoy some smallness properties. Additionally thezj should converge in some sense as t→∞ where only one zj converges toa nonzero value. The function η should scatter and converges to a solutionof the free linear Schro¨dinger equation. While we would like to, after under-standing the existence on nonlinear bound states and the linearized operator,proceed as in [42], [59], [27] we draw attention to the fact that eigenvaluesarbitrarily close to the threshold will adversely affect the necessary decay914.2. Small Solutions to the Gross-Pitaevskii Equationestimates. 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