{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Mathematics, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Chang, Yifan","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2014-07-23T15:12:06Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"2014","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Science - MSc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"In this thesis, we construct spot equilibrium asymptotic solutions to the Bruusselator model in the semi-strong interaction regime characterized by an asymptotically large diffusivity ratio under two settings: periodic solutions in R\u00b2 with respect to a Bravais lattice and spot solutions concentrate around some discrete points inside a finite domain. We use matched asymptotic methods, Bloch theory and the study of certain nonlocal eigenvalue problems to do the stability analysis of the linearised system and calculate the two term asymptotic approximation for the stability threshold. In the end we compare the numerical results with the asymptotic approximations, use Ewald\u2019s methods to derive an explicit expression for the regular part of the Bloch Green function to decide the optimal lattice arrangement and do a case study for the N-peak solutions on a ring inside the unite disk.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/48476?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"The Stability of Spot Patterns for theBrusselator Reaction-Diffusion Systemin Two Space Dimensions:Periodic and Finite Domain SettingsbyChang, YifanB.Sc., Peking University, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2014c\u00a9 Chang, Yifan 2014AbstractIn this thesis, we asymptotically construct steady-state localized spot solu-tions to the Brusselator reaction-diffusion system in the semi-strong inter-action regime characterized by an asymptotically large diffusivity ratio. Weconsider two distinct settings: a periodic pattern of localized spots in R2 con-centrating at lattice points of a Bravais lattice, and multi-spot solutions thatconcentrate around some discrete points inside a finite domain. We use themethod of matched asymptotic expansions, Floquet-Bloch theory, and thestudy of certain nonlocal eigenvalue problems to perform a linear stabilityanalysis of these patterns. This analysis leads to a two-term approximationfor a certain stability threshold characterized by a zero-eigenvalue crossing.Numerical results for the stability threshold are obtained, and comparedwith various approximations. For the periodic problem, a key feature forthe determination of the stability threshold is to use an Ewald summationmethod to derive an explicit expression for the regular part of the BlochGreen function. Moreover, such an expression allows for the identificationof the lattice that offers the optimum stability threshold. For the finitedomain problem, we implement our asymptotic theory by calculating thestability threshold for an N -spot pattern where the spots are equidistantlyspaced on a circular ring that is concentric within the unit disk.iiPrefaceThis thesis is original, unpublished, independent work by the author, Y.Chang under the supervision Dr. M. Ward and Dr. J. Wei.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Preliminaries: The Bravais Lattice and the Bloch Green\u2019sFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Bravais Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Bloch Theorem and Bloch Green Function . . . . . . . . . . 123 Periodic Spot Patterns for the Brusselator . . . . . . . . . . 193.1 Periodic Spot Solutions . . . . . . . . . . . . . . . . . . . . . 193.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 263.3 A Quick Derivation of the Stability Threshold . . . . . . . . 354 Spot Patterns for the Brusselator on a Finite Domain . . 384.1 The N -Spot Solutions . . . . . . . . . . . . . . . . . . . . . . 394.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 43ivTable of Contents4.2.1 \u03bb 6= 0 and \u03bb \u223c O(1) . . . . . . . . . . . . . . . . . . . 444.2.2 \u03bb \u223c O(\u03bd) and \u03bb 6= 0 . . . . . . . . . . . . . . . . . . . 484.2.3 \u03bb=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 555.1 Small S Asymptotics of \u03c7(S, f) . . . . . . . . . . . . . . . . 555.2 Stability Threshold and the Optimal Lattice Arrangement . 575.3 Case Study: N Peaks on a Ring . . . . . . . . . . . . . . . . 656 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71vList of Tables5.1 Source strength threshold and its asymptotic approximationfor a regular hexagonal lattice with |\u2126| = 1 and f = 0.4. . . . 655.2 The stability threshold in terms of the source strength S andits one- and two-term asymptotic approximation for a 5 spotpattern on a ring of radius 0.5 concentric within the unit diskwith f = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66viList of Figures2.1 Wigner-Seitz primitive cell . . . . . . . . . . . . . . . . . . . . 83.1 The Continuous Band of Spectra . . . . . . . . . . . . . . . . 345.1 Numerical solution (bottom curves) and asymptotic results(top curves) for \u03c7(S, f). In the left panel we fix f = 0.4,while f = 0.5 for the right panel. In both pictures, the blue(bottom) curve is the numerical solution while the green (top)one is the two term asymptotic expansion. . . . . . . . . . . . 575.2 Numerical solution to (5.5) and the two-term asymptotic ap-proximations for Sc with different c. Left panel: f = 0.4 and\u000f = 0.01. Right panel: f = 0.5 and \u000f = 0.05. The blue (top)curve is the numerical solution while the green (bottom) oneis the asymptotic approximations in both cases. . . . . . . . . 595.3 5 localized spots on a ring concentric within the unit disk. . . 65viiAcknowledgementsI would like to thank Dr. Michael Ward and Dr. Juncheng Wei for theirgreat guidance and generous support. I also want to thank all my friendshere at UBC who have made these two years a wonderful experience for me.In the end I want to thank my parents for their love through all these years.viiiChapter 1IntroductionInspired by the work of Allan Turing [14] in 1952, there has been much ef-fort over the past five decades in trying to characterize various patterns thatappear in the physical world through the modeling and analysis of reaction-diffusion (RD) systems. In [14], Turing proposed a mechanism for biologicalmorphogenesis based on his analytical study of a coupled two-componentsystem of reaction-diffusion equations with very different diffusion coeffi-cients. Using a linear stability analysis, he found that a small perturbationto a spatially homogeneous initial data can develop into certain spatial pat-terns through an instability. In the current literature, this type of instabilityis now referred to as a Turing instability. Since then, various RD systemshave been proposed and analyzed to model both biological and chemicalpatterns. Such systems include the Gray-Scott model (cf. [7]), and theGierer-Meinhardt system (cf [6]).Spatially localized spot patterns have been observed both in chemicaland numerical experiments. A survey of such patterns is given in [15].Over the past decade there has been a considerable focus on developinga theoretical mathematical framework for the study of localized patterns forsingularly perturbed reaction diffusion systems for which the ratio O(\u000f\u22122) ofthe two diffusivities is asymptotically large. The work of [17] gives a reviewon the existence, classification and stability of multiple-peaked solutionsfor the Gierer-Meinhardt system on an interval I \u2282 R1. The study [16]generalizes the results to spike solutions in a finite domain \u2126 \u2282 R2. In bothpapers, the existence of the multi-peaked solutions is proved rigorously byusing the Lyapunov-Schmidt reduction method, while the stability results onthe so-called large eigenvalues with \u03bb = O(1) as \u000f\u2192 0 are based on the studyof certain nonlocal eigenvalue problems. In [3] formal singular perturbation1Chapter 1. Introductiontechniques, based on the method of matched asymptotic expansions, areused to analyze the stability of spot patterns for the 2-D Gray-Scott system.In this work, the slow dynamics of the spot patterns is also characterized.The study [10] analyzes the self-replication process of spot patterns for theSchnakenburg model. A formal asymptotic analysis is used to derive an ODEsystem describing the slow dynamics of the spot patterns, which has thesame effect of summing infinite logarithmic series in powers of \u03bd = \u22121\/ ln \u000f.In this thesis, we construct localized spot solutions to the 2-D Brusselatorin the semi-strong interaction regime characterized by an asymptoticallylarge diffusivity ratio. We then study the stability of these localized patterns.The Brusselator, proposed by Prigogine and co-workers in Brussels in 1960s,is a theoretical model for a hypothetical autocatalytic reaction (cf. [12]). Thestandard form of this model can be written in terms of non-dimensional spacevariables asUT = \u000f20\u2206U + E \u2212 (B + 1)U + U2V , VT = D\u2206V +BU \u2212 U2V , (1.1)where \u000f20 = Du\/L2, D = Dv\/L2, L is a characteristic length-scale, while Duand Dv are the diffusivities of U and V . Many different patterns have beenobserved for this model through full scale numerical studies of the PDE, andvia Turing-type stability analysis augmented by weakly nonlinear theoriesfor the evolution of small amplitude patterns.Our goal in this thesis is to construct localized spot solutions underthe singularly perturbed limit \u000f0 \u2192 0 when E = O(\u000f0), while maintainingan asymptotically large diffusivity ratio Dv\/Du = O(\u000f\u221220 ), so that D =Dv\/L2 = (Dv\/Du)(Du\/L2) = O(1). Then, upon writing E = \u000f0E0 withE0 = O(1), T = t\/(B + 1), U = Bu\/\u000f0, and V = \u000f0v as in [13], thesingularly perturbed RD system becomesut = \u000f2\u2206u+ \u000f2E \u2212 u+ fu2v , \u03c4vt = D\u2206v +1\u000f2(u\u2212 u2v) . (1.2)2Chapter 1. Introductionwhere we have defined f , \u03c4 , E, D, and \u000f, byf \u2261BB + 1, \u03c4 \u22611f2, E \u2261E0B, D \u2261D(B + 1)B2, \u000f \u2261\u000f0\u221aB + 1,This system has three independent bifurcation parameters E, D, and f ,depending on D, B, and E0. From the definition of f , we observe thatf \u2208 (0, 1).In order to solve the system (1.2), we still need some information aboutthe domain and boundary conditions. We will consider two cases in thisthesis; periodic solutions on R2 and multi-spot solutions on a finite domain.For the periodic case, we look for periodic solutions to (1.2) with respectto some Bravais lattices, i.e.u(x+ li) = u(x), v(x+ li) = v(x), i = 1, 2, (1.3)where li, i = 1, 2 are two Bravais vectors. The study [8] undertakes a similaranalysis for the Gierer-Meinhardt and Schnakenburg reaction-diffusion sys-tems. Some basic facts concerning Bravais lattices, the Wigner-Seitz prim-itive cell, reciprocal lattices and Brillouin zones are reviewed in Chapter 2.Due to the periodicity, we can avoid solving the system on the whole plane.Instead, to construct an equilibrium, or steady-state solution, we need onlyconsider a Wigner-Seitz primitive cell together with periodic boundary con-ditions on the boundary of the cell. After constructing periodic solutions inthe primitive cell using the method of matched asymptotic expansions, weperturb this solution and perform a linear stability analysis. The solutionswe construct are stable for small D, unstable for D large, and the stabilitythreshold occurs when D \u223c O(ln 1\u000f ). To calculate the stability threshold, weneed to find the spectrum of the singularly perturbed eigenvalue problemon the whole plane, which is equivalent to finding the eigenfunction of theoperator \u2206 + V (x), where V (x) is a periodic function with respect to theBravais lattices. In this way, the Floquet-Bloch theory arises naturally inthe formulation of the stability problem. We prove some basic facts regard-ing the Bloch Green function in Section 2.3. According to the Floquet-Bloch3Chapter 1. Introductiontheory, instead of solving the perturbed system on the whole plane, we onlyneed to solve it within the the Wigner-Seitz primitive cell with the Blochquasi-periodic boundary conditions, which involve a Bloch vector k (notunique) in the first Brillouin zone. By analyzing this problem using themethod of matched asymptotic expansions, we derive a nonlocal eigenvalueproblem. From this problem we obtain the leading order result for the sta-bility threshold, which is independent of the geometry of the lattice andthe Bloch vector. To determine a higher-order approximation for the stabil-ity threshold, we perform a more refined perturbation analysis in order tocalculate a real-valued band of continuous spectrum lying within an O(\u03bd)neighbourhood of the origin in the spectral plane. This band does dependon the geometry of the lattice and the Bloch vector k. For a given lattice,we determine the next term in the stability threshold from the requirementthat the rightmost edge of the real-valued continuous band of spectrum liein the left plane {\u03bb|Re(\u03bb) \u2264 0} for any Bloch vector k in the first Brillouinzone. This calculation involves minimizing the regular part of the BlochGreen\u2019s function. Then, we can determine the optimal lattice arrangementwhich allows for the largest stability threshold. Overall, the identificationof the optimal stability threshold is through a min-max argument. Finally,in addition to this detailed way to calculate the stability threshold, whichinvolves calculating a continuous band of spectrum near the origin of thespectral plane, we also give a quick, but formal, alternative derivation of thefirst two terms of the stability threshold in Section 3.3.Our second problem concerns the analysis of the multi-spot patterns to(1.2) on a finite domain \u2126 \u2282 R2 with the no-flux boundary conditions:\u2202nu(x) = 0 , \u2202nv(x) = 0 x \u2208 \u2202\u2126 . (1.4)Here n is the outer normal vector on the boundary. We focus on both theexistence and linear stability of multi-spot patterns for this problem. Thestudy [13] analyzes a similar problem on a sphere for the Brusselator. Weconstruct asymptotic spot solutions whose u component concentrates on Ngiven points x1,x2, ...,xn \u2208 \u2126. For simplicity, we will only consider the case4Chapter 1. Introductionfor which these N spots have a common height. As shown in [16] for therelated Gierer-Meinhardt system, the true steady-state positions of theseN points are not arbitrary but rather are close to the critical point of acertain objective function related to the Neumann Green function. It isanticipated that a similar result should hold for the true steady-state spotlocations for the Brusselator. In our approach, we focus not only on steady-state solutions, but also on quasi steady-state solutions that can persistover very long time intervals. Our only key assumption is that the spotpattern has sufficient symmetry so that the vector e = (1, 1, ..., 1)T is aneigenvector of a certain Neumann Green\u2019s matrix. When this requirementis met, there is a common local behavior near each of the spots. Afterconstructing asymptotic solutions that have this symmetry, we introduce aperturbation and perform a linear stability analysis of these patterns as forthe periodic case. The key difference between the periodic and finite domainproblems is that, instead of solving the system only within one primitivecell in the periodic case, the interaction of the spots for the finite domainproblem arises through a Neumann Green matrix. A detailed calculation ofthe stability threshold for this problem, based on the method of matchedasymptotic expansions, involves the eigenvalues and eigenfunctions of theNeumann Green matrix in an essential way.The last chapter of this thesis is concerned with performing a few nu-merical experiments. Firstly, we numerically calculate a key term \u03c7(S, f),which appears in the boundary conditions when solving the inner core prob-lems near a spot. The numerical results for this quantity are comparedwith asymptotic approximations for it that are derived in the small sourcestrength limit S \u001c 1. Then we calculate the stability threshold directlyfrom a nonlinear algebraic equation, and we compare the results with one-and two-term asymptotic approximations. For the periodic problem, we alsoshow that a regular hexagonal lattice of spots offers the optimum stabilitythreshold. Finally, we consider a case study for the finite domain problem,in which N spots are equidistantly-spaced on a circular ring that is concen-tric with the unit disk. For this configuration, a refined approximation forthe stability threshold is calculated.5Chapter 2Preliminaries: The BravaisLattice and the BlochGreen\u2019s Function2.1 Bravais LatticeIn this section we review some basic definitions and results regarding theBravais lattice. The Bravais lattice was first introduced to describe theperiodicity of crystalline solids. The lattice points of a Bravais lattice in Rdcan be represented as\u039b = {d\u2211i=1nili|ni \u2208 Z, li are linear independent in Rd, i = 1, 2, ..., n} ,(2.1)so that the lattice looks exactly the same when translated by any integer lin-ear combination of li, which are called lattice vectors. We remark here thatthe choice of lattice vectors is not unique. In fact, any linear transformation{\u2211nj=1 aijlj}ni=1 with det(aij) = \u00b11 will give the same lattice.Definition 2.1.1. A primitive cell of a Bravais lattice is the smallest regionwhich when translated by all different integer linear combinations of latticevectors can cover the entire space without overlapping. The Wigner-Seitzprimitive cell of a lattice point is a special primitive cell consisting of allpoints in space that are closer to this lattice point than to any other latticepoint.As we can see from the definition, the Wigner-Seitz primitive cell of a62.1. Bravais Latticecertain lattice point is unique while a primitive cell is not. We can determinethe Wigner-Seitz primitive cell of a certain lattice point by identifying thesmallest region enclosed by all hyperplanes which perpendicularly bisect theline segments between this point and any other lattice point (it is sufficientto only consider the nearby points around the chosen point).Now we consider the lattice on R2. A Bravais lattice \u039b on R2 is generatedby two lattice vectors l1, l2 which are not parallel to each other. Withoutlose of generality, we may assume l1 is (1, 0), and we can always choosel2 = (a, b), such that a \u2208 (0, 1], b > 0 by adding some kl1, for k \u2208 Z, to theoriginal l2. Then we consider the Wigner-Seitz primitive cell of the origin Oand show that it is either a hexagon or a rectangle. First of all, due to thecentral symmetry of the lattice, we know that the boundaries of the Wigner-Seitz primitive cell will come in pairs and so we only need to look at theright-hand side. Next, after some easy observation, it follows that b1 and b2,which is the perpendicular bisector of \u00b1l2, form part of the boundaries of theWigner-Seitz primitive cell. The remainder of the boundary arises from someof the perpendicular bisectors of the line segments between the origin and thepoints of the second column, i.e. l1 + kl2, k \u2208 Z. It is useful to observe themid-points of these line segments lie on a line parallel to l2 passing throughthe mid-point of OA and the distance between two mid-points next to eachother is |l2|2 . Since b1, b2 are perpendicular to l2, and the distance betweenthem is |l2|, then for most of the cases only two of these mid-points will liebetween them and the corresponding perpendicular bisectors intersectingwith b1, b2 form the boundaries of the primitive cell. This is the genericcase when the Wigner-Seitz primitive cell is a hexagon. A special case iswhen one mid-point lies in the middle of b1, b2 and the two mid-points nextto it on b1, b2, Then, the perpendicular bisectors corresponding to thesethree mid-points is the same line which is perpendicular to b1, b2. Thus, inthis case the Wigner-Seitz primitive cell will be a rectangle. In conclusion,if we use the polar coordinate to denote the angle between l1 and l2 by \u03b8,and the length of l2 by r, then by the assumption we made on l2, we willhave \u03b8 \u2208 (0, pi2 ), r \u2208 (0,1cos \u03b8 ) and the Wigner-Seitz primitive cell will be ahexagon unless r = cos \u03b8k , k \u2208 Z, k > 0. In this latter, degenerate case, the72.2. Reciprocal Latticeprimitive cell will be a rectangle.\u22121.5 \u22121 \u22120.5 0 0.5 1 1.5 2\u22121\u22120.500.511.5Figure 2.1: Wigner-Seitz primitive cell2.2 Reciprocal LatticeNext, we review the concept of the reciprocal lattice \u039b\u2217 of a certain Bravaislattice. This notion arises from Fourier analysis. Firstly, we define a periodicfunction with respect to a Bravais lattice as follows:Definition 2.2.1. Given a Bravais lattice \u039b in Rd with lattice vectors{li}di=1, a function u(x) is periodic with respect to the lattice \u039b if u(x+li) =u(x), i = 1, 2, ..., d.We know that a 2pi periodic function u(x) on R can be decomposed intoFourier series as einx, n \u2208 Z. According to Definition 2.2.1, we can viewu(x) as a periodic function with respect to the 1-D lattice \u039b = {nl1|n \u2208Z, l1 = 2pi} and the Fourier basis einx, n \u2208 Z can be viewed as eik\u00b7x, k \u2208\u039b\u2217 = {nl\u22171|n \u2208 Z, l\u22171 = 1}, where \u039b\u2217 is another Bravais lattice relatedto \u039b, which will be defined as the reciprocal lattice of \u039b later. In higherdimensions, we first consider a 2pi periodic (in each direction) function u(x),i.e. u(x + 2piei) = u(x), i = 1, 2, ..., d, where ei are the standard basis ofRd. We know that u(x) can be decomposed into Fourier series of ein\u00b7x,n \u2208 Zd. As explained above, u(x) now can be viewed as a periodic functionwith respect to the Bravais lattice \u039b = {\u2211di=1 nili|ni \u2208 Z, li = 2piei, i =1, 2, ..., d}. In addition, the Fourier basis ein\u00b7x, n \u2208 Zd can be viewed as82.2. Reciprocal Latticeeik\u00b7x, k \u2208 \u039b\u2217 = {\u2211di=1 nil\u2217i |ni \u2208 Z, l\u2217i = ei}. For the general case, if u(x) isa periodic function with respect to a Bravais lattice \u039b with lattice vectors{li}di=1, we want to decompose it as we did above. In order to convert to theprevious case, we use a linear coordinate change x = 12pi [l1|...|ld]y and denoteA = 12pi [l1|...|ld]. Then u\u02dc(y) = u(Ay) is a 2pi periodic function in each yidirection and can be decomposed into Fourier series as ein\u00b7y = ein\u00b7(A\u22121x) =ei((A\u22121)Tn)\u00b7x, n \u2208 Zd. If we denote (A\u22121)T = [b1|...|bd], then (A\u22121)Tn,n \u2208 Zd, represents a Bravais lattice with lattice vector {bi}d1. And sinceAT (A\u22121)T = I, we have the relation ( 12pi li) \u00b7 bj = \u03b4ij , so that li \u00b7 bj = 2pi\u03b4ij ,which leads to the definition of reciprocal lattice.Definition 2.2.2. The reciprocal lattice \u039b\u2217 of a Bravais lattice \u039b with latticevectors {li}di=1 is the Bravais lattice given by the lattice vectors {bi}di=1 wherebi are the vectors satisfy li \u00b7 bj = 2pi\u03b4ij.We remark that { 12pibi}di=1 are the dual basis for {li}di=1 in Rd. Thus,{bi}di=1 are well defined and uniquely determined by {li}di=1. We furtherremark that other authors choose the reciprocal lattice vectors {bi}di=1 tosatisfy li \u00b7 bj = \u03b4ij .Some important properties concerning reciprocal lattices \u039b\u2217 of a Bravaislattice \u039b are listed here as follows:\u2022 \u039b\u2217 = {b \u2208 Rd|eib\u00b7l = 1, \u2200l \u2208 \u039b}\u2022 The reciprocal lattice of \u039b\u2217 is \u039b.\u2022 Any periodic function u(x) with respect to \u039b can be decomposed intoFourier series of eib\u00b7x, where b \u2208 \u039b\u2217. Using a change of variable, wehave the formula:u(x) =1|\u2126|\u2211b\u2208\u039b\u2217(\u222b\u2126u(y)e\u2212ib\u00b7y dy)eib\u00b7x . (2.2)\u2022 The Fourier transform of\u2211l\u2208\u039b \u03b4(x\u2212 l) is(2pi)d|\u2126|\u2211b\u2208\u039b\u2217 \u03b4(\u03be\u2212 b), where\u2126 is the Wigner-Seitz primitive cell of \u039b.92.2. Reciprocal Lattice\u2022 If u(x) \u2208 L1(Rd), then\u2211l\u2208\u039b u(x + l) converges absolutely almosteverywhere, and we have:\u2211l\u2208\u039bu(x+ l) =1|\u2126|\u2211b\u2208\u039b\u2217u\u02c6(b)eix\u00b7b . (2.3)In particular, upon replacing u(x) by u(x)eix\u00b7k, we have that\u2211l\u2208\u039bu(x+ l)eik\u00b7l =1|\u2126|\u2211b\u2208\u039b\u2217u\u02c6(b\u2212 k)eix\u00b7(b\u2212k) . (2.4)We remark that the last two properties are called Poisson summation for-mulae. We prove these properties as follows:Proof. First we establish the basic result that the Fourier transform of\u2211l\u2208\u039b \u03b4(x \u2212 l) is(2pi)dV\u2211b\u2208\u039b\u2217 \u03b4(\u03be \u2212 b). The key step in the proof is thefollowing equality:\u221e\u2211n=\u2212\u221eeinx = 2pi\u221e\u2211k=\u2212\u221e\u03b4(x\u2212 2pik) . (2.5)The proof of (2.5) can be found in many books; we simply sketch the outlineof the proof here. First of all, the equality holds in the sense of distribution,and since both sides are 2pi periodic, we may prove it only on the interval[\u2212pi, pi], i.e. for any test function \u03c6(x) \u2208 C\u221e0 [\u2212pi, pi],limN\u2192\u221e\u222b pi\u2212pi(N\u2211n=\u2212Neinx)\u03c6(x) dx =\u222b pi\u2212pi(2pi+\u221e\u2211k=\u2212\u221e\u03b4(x\u2212 2pik))\u03c6(x) dx = 2pi\u03c6(0) .(2.6)This is true sinceN\u2211n=\u2212Neinx =sin (N + 12x)sin x2\u2192 \u03b4(x) , as N \u2192\u221e ,in the sense of distributions.102.2. Reciprocal LatticeNotice that the 1-D case followed directly from this equality,\u0302\u221e\u2211n=\u2212\u221e\u03b4(x\u2212 nl) =\u221e\u2211n=\u2212\u221eein\u00b7\u03bel = 2pi\u221e\u2211k=\u2212\u221e\u03b4(l\u03be\u22122pik) =2pil\u221e\u2211k=\u2212\u221e\u03b4(\u03be\u2212k2pil) .For the higher dimensional case, we have\u0302\u2211l\u2208\u039b\u03b4(x\u2212 l) =\u2211mi\u2208Ze\u2212i(\u2211di=1mili)\u00b7\u03be =d\u220fi=1(\u221e\u2211mi=\u2212\u221ee\u2212imi(li\u00b7\u03be)) (2.7)=d\u220fi=1(2pi\u221e\u2211ki=\u2212\u221e\u03b4(li \u00b7 \u03be \u2212 2piki)) =d\u220fi=1(\u221e\u2211ki=\u2212\u221e\u03b4(li \u00b7 \u03be2pi\u2212 ki)) .If we denote \u03be =\u2211di=1 \u03b7ibi = [b1|b1|...|bd]\u03b7 = B\u03b7, where {bi}di=1 are thereciprocal vectors to {li}di=1, then (2.7) becomesd\u220fi=1(\u221e\u2211ki=\u2212\u221e\u03b4(li \u00b7 \u03be2pi\u2212 ki)) =d\u220fi=1(\u221e\u2211ki=\u2212\u221e\u03b4(\u03b7i \u2212 ki)) =\u2211k\u2208Zd\u03b4(\u03b7 \u2212 k) (2.8)=\u2211k\u2208Zd\u03b4(B\u22121(\u03be \u2212Bk)) =1det(B\u22121)\u2211b\u2208\u039b\u2217\u03b4(\u03be \u2212 b) .SinceB\u22121 = 12pi [l1|l1|...|ld], then det(B\u22121) = V(2pi)d , where V = det([l1|l1|...|ld])is the volume of the primitive cell. In this way, we conclude that\u0302\u2211l\u2208\u039b\u03b4(x\u2212 l) =(2pi)dV\u2211b\u2208\u039b\u2217\u03b4(\u03be \u2212 b) . (2.9)Then we prove the last property. First of all since\u222b\u2126\u2211l\u2208\u039b|u(x+ l)| dx =\u222bR2|u(x)| dx <\u221e , (2.10)the series\u2211l\u2208\u039b u(x+ l) converges absolutely almost everywhere. Moreover,since the series is periodic with respect to the lattice \u039b, then by using the112.3. Bloch Theorem and Bloch Green Functionproperty above we can decompose it into a Fourier series as\u2211l\u2208\u039bu(x+ l) =1|\u2126|\u2211b\u2208\u039b\u2217(\u222b\u2126(\u2211l\u2208\u039bu(y + l))e\u2212ib\u00b7y dy)eib\u00b7x . (2.11)The last step in the derivation is to calculate the Fourier coefficients as\u222b\u2126(\u2211l\u2208\u039bu(y + l))e\u2212ib\u00b7y dy =\u2211l\u2208\u039b\u222b\u2126u(y + l)e\u2212ib\u00b7y dy (2.12)=\u222bR2u(y)e\u2212ib\u00b7y dy = u\u02c6(b) , (2.13)where the the second to last equality holds since R2 is tiled when \u2126 trans-lated by all lattice vector, and since \u2200l \u2208 \u039b and \u2200b \u2208 \u039b\u2217, we have eil\u00b7b = 1.In particular when we replace u(x) by u(x)eix\u00b7k, the Fourier transformis translated by k.There are two further useful concepts that relate to Bravais lattices.Definition 2.2.3. Bragg planes are the hyperplanes which perpendicularlybisect any line segment between two lattice points of \u039b\u2217. The first BrillouinZone is the Wigner-Seitz primitive cell of \u039b\u22172.3 Bloch Theorem and Bloch Green FunctionIn this section, we review Bloch theorem and some properties of the BlochGreen function in the Wigner-Seitz primitive cell \u2126 of some lattice \u039b. Theseresults will be used later when we consider the stability of a periodic patternof spots for the reaction-diffusion system. The proof below is similar to thatin [8].The Bloch theorem originates from quantum mechanics and states thatthe eigenfunction \u03c6(x) of the operator \u2212\u2206 + V (x), where the potentialfunction V (x) is periodic with respect to a Bravais lattice \u039b, must havethe form \u03c6(x) = eik\u00b7x\u03c6p(x), where \u03c6p(x) is also periodic with respect tothe lattice \u039b, and k can be chosen to lie in the first Brillouin Zone, orequivalently \u2200l \u2208 \u039b, \u03c6(x + l) = eik\u00b7l\u03c6(x). The Bloch theorem allows us122.3. Bloch Theorem and Bloch Green Functionto solve the eigenvalue problem within the primitive cell together with theBloch boundary conditions, instead of on the whole space. The proof ofthe Bloch theorem can be found in many solid physics books and the keyidea is that if two operators commute, which in this case are \u2212\u2206 + V (x)and translation by any lattice vector, they share common non-degenerateeigenvectors while the eigenvalues may be different. By using this idea, wecan prove the Bloch theorem for a system, which is the basis for our linearstability analysis. Since the equilibrium solutions we construct are periodicwith respect to some Bravais lattice \u039b, finding the eigenspace of the linearperturbed system is equivalent to finding the eigenfunction for an operatorsimilar to \u2212\u2206 + V (x). In this way, the Bloch theorem arises naturally inthe stability analysis of a periodic arrangement of spots.Next we consider some key properties of the Bloch Green function. Thisis the Green function in the fundamental Wigner-Seitz cell that satisfies theBloch boundary conditions. As we have shown above, the primitive cell \u2126will be either a hexagon or a rectangle. We may assume that the boundariesof \u2126 consist of d\u00b1i, with i \u2264 2 for a rectangle and i \u2264 3 for a hexagon, whered\u00b1i represents the perpendicular bisector of \u00b1Li \u2208 \u039b which come in pairs.The Bloch Green function in \u2126 is the solution to\u2206G0,k(x) = \u2212\u03b4(x), x \u2208 \u2126, (2.14)and satisfies the following Bloch boundary conditions, also referred to asquasi-periodic boundary conditions:\u2200x \u2208 d\u2212i, G0,k(x+Li) = e\u2212ik\u00b7LiG0,k(x), (2.15)\u2202n\u2212G0,k(x+Li) = e\u2212ik\u00b7Li\u2202\u2212n+G0,k(x),where the \u00b1 behind n in the directional derivatives denote one-side deriva-tives, n is the outer unit normal vector parallel to Li and k is some non-zerovector in the first Brillouin Zone \u2126\u2217, i.e. in the Wigner-Seitz primitive cellof \u039b\u2217. We first remark here that we require k 6= 0 since there is no solutionto (2.14) if k = 0. This is shown by integrating \u2206G0,k(x) over \u2126, and us-132.3. Bloch Theorem and Bloch Green Functioning the divergence theorem which results in a contradiction. The boundaryconditions are well-defined since x + Li \u2208 di if x \u2208 d\u2212i. Moreover, we useone-side normal derivative since we only solve the equation inside \u2126. Withthis boundary conditions, we can extend the solution to the whole planewith continuous normal derivative between contiguous cells.It is also useful to analyze the quasi-periodic reduced-wave Green\u2019s func-tion, which is the solution to\u2206G\u03c3,k(x)\u2212 \u03c32G\u03c3,k(x) = \u2212\u03b4(x) , x \u2208 \u2126 , \u03c3 \u2208 R , (2.16)with the boundary conditions of (2.15). We observe that the Bloch Greenfunction is simply the special case of the reduced-wave Green\u2019s functionwhen \u03c3 = 0. The first key property is that the regular part R\u03c3,k of G\u03c3,k(x),which is defined by subtracting the free space Green function \u2212 12pi ln |x| fromG\u03c3,k(x), i.e.R\u03c3,k = limx\u21920G\u03c3,k(x) +12piln |x| ,is real for k 6= 0.Lemma 1. The regular part R\u03c3,k of G\u03c3,k(x) is real-valued for |k| 6= 0.Proof. We let \u000f > 0 and eliminate the singularity by cutting a small ballB(0, \u000f) around the origin. We denote \u2126\u000f \u2261 \u2126 \u2212 B(0, \u000f). Then we use thedivergence theorem and the fact that \u2206G\u03c3,k(x) = \u03c32G\u03c3,k(x) for x \u2208 \u2126\u000f tocalculate\u222b\u2202\u2126\u000fG\u03c3,k\u2202nG\u03c3,k dl =\u222b\u2202\u2126\u000fG\u03c3,k(\u2207G\u03c3,k \u00b7 n) dl =\u222b\u2202\u2126\u000f((G\u03c3,k\u2207G\u03c3,k) \u00b7 (n dl)=\u222b\u2126\u000f\u2207 \u00b7 (G\u03c3,k\u2207G\u03c3,k) dx =\u222b\u2126\u000f(\u2207G\u03c3,k \u00b7 \u2207G\u03c3,k +G\u03c3,k\u2206G\u03c3,k)dx=\u222b\u2126\u000f(|\u2207G\u03c3,k|2 + \u03c32|G\u03c3,k|2) dx ,where n is the outer normal vector as usual. Upon calculating the boundary142.3. Bloch Theorem and Bloch Green Functionintegral directly, we have\u222b\u2202\u2126\u000fG\u03c3,k\u2202nG\u03c3,k dl =\u222b\u2202\u2126G\u03c3,k\u2202nG\u03c3,k dl \u2212\u222b\u2202B(0,\u000f)G\u03c3,k\u2202nG\u03c3,k dl . (2.17)We claim that the first term vanishes due to the Bloch boundary conditions(2.15). Since \u2202\u2126 has either 4 or 6 perpendicular bisectors, which come inpairs, then according to (2.15), on each pair d\u00b1i, G\u03c3,k(x) is different by afactor of e\u2212ik\u00b7Li and \u2202nG\u03c3,k is different by \u2212e\u2212ik\u00b7Li owing to the fact thatthe outer normal vectors are opposite. Then, if we integrate G\u03c3,k\u2202nG\u03c3,k ond\u00b1i, these two terms will be different by e\u2212ik\u00b7Li(\u2212e\u2212ik\u00b7Li) = \u22121. Thus, thefirst integral on the right hand-side of (2.17) banishes. Next, we calculatethe second term on the right hand-side of (2.17) as\u222b\u2202B(0,\u000f)G\u03c3,k\u2202xG\u03c3,k dx \u223c \u000f\u222b 2pi0(\u221212piln \u000f+R\u03c3,k + o(1))(\u221212pi\u000f+O(1)) d\u03b8\u223c12piln \u000f\u2212R\u03c3,k +O(\u000f ln \u000f) .In this way, we let \u000f\u2192 0 to obtainR\u03c3,k = lim\u000f\u21920[ \u222b\u2126\u000f(|\u2207G\u03c3,k|2 + \u03c32|G\u03c3,k|2) dx+12piln \u000f], (2.18)which proves that R\u03c3,k is real-valued.As we mentioned before, there is no solution to (2.14) if k = 0. So thenext lemma will discuss the asymptotic behaviour of R0,k when k tends to0. To analyze this limiting behavior, we may assume k = \u03c3\u03ba, where \u03c3 \u001c 1and |\u03ba| = O(1). This form suggests that we can calculate a solution to(2.14) by a singular perturbation technique. As such, we expand G0,k asG0,k(x) =U0(x)\u03c32+U1(x)\u03c3+ U2(x) + \u00b7 \u00b7 \u00b7 . (2.19)In addition, the expansion of the boundary conditions (2.15) yield for \u2200x \u2208152.3. Bloch Theorem and Bloch Green Functiond\u2212i thatU0\u03c32+U1\u03c3+ ...\u2223\u2223x+Li= [1\u2212 i\u03c3(\u03ba \u00b7Li) + ...](U0\u03c32+U1\u03c3+ ...)\u2223\u2223x\u2202n\u2212U0\u03c32+\u2202n\u2212U1\u03c3+ ...\u2223\u2223x+Li= \u2212 [1\u2212 i\u03c3(\u03ba \u00b7Li) + ...](\u2202\u2212n+U0\u03c32+\u2202\u2212n+U1\u03c3+ ...)\u2223\u2223x,where n is the outer normal vector on the boundary, which is parallel to Li.Then by equating the same order of \u03c3, we get:O(\u000f\u22122) : \u2206U0(x) = 0 , \u2200x \u2208 \u2126 ,\u2200x \u2208 d\u2212i , U0(x+Li) = U0(x) , \u2202n\u2212U0(x+Li) = \u2202\u2212n+U0(x) .O(\u000f\u22121) : \u2206U1(x) = 0 , \u2200x \u2208 \u2126 ,\u2200x \u2208 d\u2212i , U1\u2223\u2223x+Li= U1 \u2212 i (\u03ba \u00b7Li)U0\u2223\u2223x ,\u2202n\u2212U1\u2223\u2223x+Li= \u2202\u2212n+U1 \u2212 i (\u03ba \u00b7Li) \u2202\u2212n+U0\u2223\u2223x ,O(1) : \u2206U2(x) = \u2212\u03b4(x) , \u2200x \u2208 \u2126 ,\u2200x \u2208 d\u2212i , U2\u2223\u2223x+Li= U2 \u2212 i (\u03ba \u00b7Li)U1 \u2212(\u03ba \u00b7Li)22U0\u2223\u2223x ,\u2202n\u2212U2\u2223\u2223x+Li= \u2202\u2212n+U2 \u2212 i (\u03ba \u00b7Li) \u2202\u2212n+U1 \u2212(\u03ba \u00b7Li)22\u2202\u2212n+U0\u2223\u2223x .From the leading order equation we conclude that U0 = a for some constanta. Form the O(\u000f\u22121) equation, we get that U1 is a linear function of the formU1(x) = (\u2212ia\u03ba) \u00b7x+ b, where b is another constant. Next we integrate \u2206U2over \u2126 defined by the O(1) problem. Upon using the boundary conditionsand the expression for U0 and U1, as derived above, we obtain that\u222b\u2126\u2206U2 dx =\u222b\u2202\u2126\u2202nU2(x) dl =L\u2211j=1\u222b\u00b1dj\u2202nU2(x) dl =\u222b\u2126\u2212\u03b4(x) dx , \u21d2L\u2211j=1\u2212i(\u03ba \u00b7Lj)n \u00b7 (\u2212ia\u03ba)|di| = \u2212aL\u2211j=1(\u03ba \u00b7Lj)\u03ba \u00b7Li|Li||di| = \u22121 ,where L = 2 if the primitive cell is a rectangle, and L = 3 if the primitive162.3. Bloch Theorem and Bloch Green Functioncell is hexagon. In this way, we determine a asa =1\u2211Lj=1(\u03ba \u00b7Lj)2 |di||Li|=1\u03baT (\u2211Lj=1|di||Li|LjLTj )\u03ba=1\u03baTQ\u03ba, (2.20)where we denote Q =\u2211Lj=1|di||Li|LjLTj . We remark that Q is a positivedefinite matrix related only to the lattice. This leads to our second lemma:Lemma 2. For sufficiently small k, the regular part R0,k of the Bloch Greenfunction G0,k has the asymptotic behaviourR0,k \u223c1kTQk, (2.21)where Q is a positive definite matrix related to the lattice \u039b as shown above.We remark here that when we try to find the singular perturbed solutionG0,k for k \u223c O(\u000f), the reason we expand G0,k from O(\u000f\u22122) is because it can\u2019tsatisfy the boundary conditions if we start from O(\u000f\u22121).Next we state two other similar results which will be useful later.Lemma 3. The regular part R\u03c3,k of the reduced-wave Bloch Green functionG\u03c3,k has the asymptotic behaviour R\u03c3,k \u223c R0,k + O(\u03c32), as \u03c3 \u2192 0 and|k| \u223c O(1), where R0,k is the same as above.Proof. Just expand G\u03c3,k = G0 + \u03c32G1 + .... Then, we have G0 is the BlochGreen function G0,k. And since G1 is bounded for |k| \u223c O(1), we have theasymptotic behaviour R\u03c3,k \u223c R0,k +O(\u03c32).Lemma 4. The regular part R\u03c3,k of the reduced-wave Bloch Green functionG\u03c3,k has the asymptotic behaviour R\u03c3,k \u223c 1\u03c32[|\u2126|+\u03baTQ\u03ba] , as \u03c3 \u2192 0 and |k| \u223cO(\u03c3), where Q is the same positive definite matrix determined by the lattice\u039b.Proof. The proof is basically the same as we do for R0,k, the only differenceis the equation for U2 became \u2206U2 = U0\u2212\u03b4(x). So when we integrate \u2206U2,U0|\u2126| will appear.172.3. Bloch Theorem and Bloch Green FunctionSimilar to the Bloch boundary conditions (2.15), we will also use theperiodic boundary conditions:G(x+ li) = G(x), \u2202n\u2212G(x+ li) = \u2202\u2212n+G(x), \u2200x \u2208 d\u2212i, (2.22)18Chapter 3Periodic Spot Patterns forthe BrusselatorIn this chapter we construct periodic spot solutions to (1.2) by first con-structing a single spot solution inside the Wigner-Seitz primitive cell \u2126 sub-ject to the periodic boundary conditions (2.22). We then periodically extendthis solution to the whole plane. Next, we analyze the linear stability of theequilibrium solutions. We first perturb the steady state solution, to derive asingular perturbed eigenvalue problem governing the linear stability of theperiodic pattern. From this eigenvalue problem we then provide an accu-rate calculation of the stability threshold corresponding to a zero eigenvaluecrossing. We also provide a more expedient approach to derive the stabilitythreshold.3.1 Periodic Spot SolutionsThe Brusselator reaction-diffusion model has the formu\u02dct = \u000f2\u2206u\u02dc+ \u000f2E \u2212 u\u02dc+ fu\u02dc2v\u02dc ,\u03c4 v\u02dct = D\u2206v\u02dc +1\u000f2(u\u02dc\u2212 u\u02dc2v\u02dc) .From this first equation we observe that there is a spatially homogeneousequilibrium solution for \u000f\u001c 1 with u\u02dc \u223c \u000f2E, and so we make a substitutionof the form u\u02dc = u+ \u000f2E and v\u02dc = v. Then the system becomesut = \u000f2\u2206u\u2212 u+ f(u2v + 2\u000f2Euv + \u000f4Ev) , (3.1)\u03c4vt = D\u2206v + E + \u000f\u22122(u\u2212 u2v)\u2212 2Euv \u2212 \u000f2E2v .193.1. Periodic Spot SolutionsSince the steady-state problem is singularly perturbed, we need to constructa localized asymptotic expansion solutions in the inner region near the originof the fundamental Wigner-Seitz cell. Then we use the method of matchedasymptotic expansions to match the inner solution to an appropriate outersolution.In the inner region, we use an inner coordinate and perform a scaling toeliminate the D. The inner variables y, U , and V , are defined byy = \u000f\u22121x , U(y) =u(x)\u221aD, V (y) =\u221aDv(x) . (3.2)To motivate this scaling of U and V , we remark that if we assumed u \u223c D\u03b1U ,v \u223c D\u03b2V , then from the first equation we have that \u03b1 = \u2212\u03b2 = k, while fromthe second equation we conclude that D \u00b7 O(D\u2212k) \u223c O(Dk). This leads tok = 1\/2.In the inner region, to leading order the solution is radially symmetric.As such, if we denote \u03c1 = |y| then to leading order we get the core problem\u2206\u03c1U \u2212 U + fU2V = 0 , \u2206\u03c1V + U \u2212 U2V = 0 , 0 < \u03c1 <\u221e , (3.3)where U and V are functions of \u03c1. Therefore, (3.3) is an ODE system.It will have solutions when equipped with initial conditions or appropriateboundary conditions. In this case, we add boundary conditions at \u03c1 = 0and an asymptotic condition as \u03c1\u2192\u221e:U \u2032(0) = V \u2032(0) = 0 ; U \u2192 0 , V \u223c S ln \u03c1+ \u03c7(S, f) , as \u03c1\u2192\u221e , (3.4)where S is some unknown source strength to be determined, while \u03c7(S, f) issome quantity depending on S and f that is to be computed from the coresolution.We remark here that we choose U \u2032(0) = V \u2032(0) = 0 since we are lookingfor differentiable radically symmetric solutions, U \u2192 0 at infinity since wewant localized spot solutions, and we allow V to have logarithmic growth atinfinity since the solution to \u2206V = \u2212\u03b4(y) on R2 is \u2212 12pi ln |y|. To derive an203.1. Periodic Spot Solutionsidentity for S, we integrate \u2206V over R2 to obtain2piS =\u222bR2(U2V \u2212 U) dx =\u222b 2pi0d\u03b8\u222b \u221e0(U2V \u2212 U)\u03c1 d\u03c1 ,so thatS =\u222b \u221e0(U2V \u2212 U)\u03c1 d\u03c1 . (3.5)Next, we construct the outer solution. Since u is localized, the secondequation in (3.1) reduces asymptotically to0 = D\u2206v + E +(\u222b\u2126[\u000f\u22122(u\u2212 u2v)\u2212 2Euv \u2212 \u000f2E2v] dx)\u03b4(x) . (3.6)Upon using (3.5), this equation becomesD\u2206v + E = 2piS\u221aD\u03b4(x) , x \u2208 \u2126 . (3.7)Then, if we integrate (3.7) over \u2126 and use the periodic boundary conditionson \u2202\u2126, we can calculate the source strength S asS =E|\u2126|2pi\u221aD, (3.8)so that (3.7) becomes\u2206v +ED=E|\u2126|D\u03b4(x) . (3.9)To represent the solution v, we introduce the periodic Green functionGp for \u2126 as\u2206Gp =1|\u2126|\u2212 \u03b4(x) , x \u2208 \u2126 , (3.10)with the periodic boundary conditions (2.22). Since Gp is determined onlyup to a constant, we impose that\u222b\u2126Gp dx = 0. As x \u2192 0, Gp has thesingular behaviour Gp \u223c \u2212 12pi ln |x|+Rp, where the regular part Rp can becalculated explicitly for any oblique Bravais lattice, as was shown in [4].In terms of this Green\u2019s function, we have v = \u2212E|\u2126|D Gp + c. As x\u2192 0,213.1. Periodic Spot Solutionsthe limiting behavior of v isv \u223cE|\u2126|2piDln |x| \u2212E|\u2126|DRp + c . (3.11)We now match this behavior with the inner solution. From (3.4), we obtainas |y| \u2192 \u221e thatv \u223c1\u221aD(S ln |y|) + \u03c7(s, f)) =E|\u2126|2piD(lnx+1\u03bd) +\u03c7\u221aD, (3.12)where \u03bd = \u22121ln \u000f \u001c 1. Upon matching, we identify the constant c asc =\u03c7\u221aD+1\u03bdE|\u2126|2piD+E|\u2126|DRp . (3.13)Since the stability threshold occurs when D \u223c O( 1\u03bd ), then from ourformula (3.8) for S, we conclude that S \u223c O( 1\u221aD) \u223c O(\u03bd12 ). Therefore, forS = O(\u03bd1\/2), we must look for an asymptotic solution to the core problem(3.3). To motivate the scalings for U and V , we observe from (3.12) and(3.3) that V =\u221aDv \u223c O( 1\u221aD\u03bd) \u223c O(\u03bd\u221212 ), UV \u223c O(1), which leads toU \u223c O(\u03bd12 ). Since S = O(\u03bd12 ), we need \u03c7(S, f) to be order O(\u03bd\u221212 ) to matchwith V . These formal scaling arguments motivate the following asymptoticexpansion for the solution to the core problem:U \u223c \u03bd12 (U0 + \u03bdU1 + \u03bd2U2 + \u00b7 \u00b7 \u00b7 ) , V \u223c \u03bd\u2212 12 (V0 + \u03bdV1 + \u03bd2V2 + \u00b7 \u00b7 \u00b7 ) ,\u03c7 \u223c \u03bd\u221212 (\u03c70 + \u03bd\u03c71 + \u03bd2\u03c72 + \u00b7 \u00b7 \u00b7 ) , S \u223c \u03bd12 (S0 + \u03bdS1 + \u03bd2S2 + \u00b7 \u00b7 \u00b7 ) .(3.14)Next, we substitute (3.14) into (3.3) and try to construct radially sym-metric solutions at each order. At leading order, we have\u2206U0 \u2212 U0 + fU20V0 = 0 , U\u20320(0) = 0 , U0 \u2192 0 , as |y| \u2192 \u221e ,\u2206V0 = 0 , V\u20320(0) = 0 , V0 \u2192 \u03c70 , as |y| \u2192 \u221e .(3.15)223.1. Periodic Spot SolutionsFrom this system we conclude thatU0 =\u03c9f\u03c70, V0 = \u03c70 , (3.16)where \u03c9 is the unique positive radially symmetric solution (see [5]) to\u2206\u03c9 \u2212 \u03c9 + \u03c92 = 0 , (3.17)with \u03c9(y) having exponential decay as |y| \u2192 \u221e. In addition, we have\u222b\u221e0 \u03c9(\u03c1)\u03c1d\u03c1 =\u222b\u221e0 \u03c92(\u03c1)\u03c1 d\u03c1 \u2261 b upon integrating the equation for \u03c9. Fur-ther properties of this ground-state solution are given in [5].At next order, we have that\u2206U1 \u2212 U1 + f(\u03c92f2\u03c720V1 + 2\u03c9fU1) = L0U1 +\u03c92f\u03c720V1 = 0 , (3.18)\u2206V1 +\u03c9f\u03c70\u2212\u03c92\u03c70f2= 0 , (3.19)U \u20321(0) = V\u20321(0) = 0 ; U1 \u2192 0 , V1 \u2192 S0 ln |y|+ \u03c71 , as |y| \u2192 \u221e ,where the operator L0 is defined as L0u = \u2206u\u2212 u+ 2\u03c9u. By applying thedivergence theorem to the V1 equation we obtain thatS0 =bf\u03c70\u2212bf\u03c720\u21d2 \u03c70 =1\u2212 ff2bS0. (3.20)Then the solutions to (3.18) and (3.19) can be decomposed asU1 = \u2212\u03c71f\u03c720\u03c9 \u22121f3\u03c730U1P , V1 = \u03c71 +V1Pf2\u03c70, (3.21)where U1P , V1P are the unique radially symmetric solution toL0U1P \u2212 \u03c92V1P = 0 , \u2206V1P = \u03c92 \u2212 f\u03c9 ,U \u20321P (0) = V\u20321P (0) = 0 ; V1P \u2192 (1\u2212 f)b ln |y|+ o(1) , as |y| \u2192 \u221e .(3.22)233.1. Periodic Spot SolutionsTo eliminate the f -dependence in V1P , we introduce V1Q satisfying\u2206V1Q = \u03c9 , V\u20321Q(0) = 0 ; V1Q \u2192 b ln |y|+ o(1) , as |y| \u2192 \u221e .(3.23)Then we have\u2206(V1Q \u2212 \u03c9) = \u03c9 \u2212\u2206\u03c9 = \u03c92 \u21d2 V1P = (1\u2212 f)V1Q \u2212 \u03c9 . (3.24)After substituting V1Q into (3.22) we getL0U1P = (1\u2212 f)\u03c92V1Q \u2212 \u03c93 . (3.25)This suggests that we should decompose U1P by introducing U1QI and U1QII ,which are taken to satisfyL0U1QI = \u03c93 , U \u20321QI(0) = 0 , U1QI \u2192 0 , as |y| \u2192 \u221e , (3.26)L0U1QII = \u03c92V1Q , U\u20321QII(0) = 0 , U1QII \u2192 0 , as |y| \u2192 \u221e , (3.27)so that U1P = (1\u2212 f)U1QII \u2212 U1QI .At second order we obtain thatL0U2 + f(\u03c92f2\u03c720V2 + \u03c70U21 + 2\u03c9f\u03c70U1V1) = 0 ,U \u20322(0) = 0 , U2 \u2192 0 , as |y| \u2192 \u221e ,\u2206V2 + U1 \u2212 U20V1 \u22122\u03c9fU1 = 0 ,V \u20322(0) = 0 , V2 \u2192 S1 ln |y|+ \u03c72 , as |y| \u2192 \u221e .By using the divergence theorem on the V2 equation we calculate S1 asS1 =\u222b \u221e0U20V1\u03c1 d\u03c1+2f\u222b \u221e0\u03c9U1\u03c1 d\u03c1\u2212\u222b \u221e0U1\u03c1 d\u03c1 . (3.28)243.1. Periodic Spot SolutionsNext, upon integrating (3.18) we get\u222bR2L0U1 dx+\u222bR2\u03c92f\u03c720V1 dx = 0 ,\u21d2 \u2212\u222b \u221e0U1\u03c1 d\u03c1+\u222b \u221e02\u03c9U1\u03c1 d\u03c1+\u222b \u221e0\u03c92f\u03c720V1\u03c1 d\u03c1 = 0 ,where\u222bR2 \u2206U1 dx vanishes due to the boundary conditions. Upon substi-tuting this back into (3.28), together with (3.26) and (3.27), we obtain thatf1\u2212 fS1 =\u222b \u221e0U1\u03c1 d\u03c1 =\u222b \u221e0(\u2212\u03c71f\u03c720\u03c9 \u22121f3\u03c730((1\u2212 f)U1QII \u2212 U1QI))\u03c1 d\u03c1= \u2212\u03c71bf\u03c720\u22121\u2212 ff3\u03c730\u222b \u221e0U1QII\u03c1 d\u03c1+1f3\u03c730\u222b \u221e0U1QI\u03c1 d\u03c1 .We then combine this expression with (3.21) and (3.20) to calculate \u03c71 as\u03c71 =f\u03c720b(\u22121\u2212 ff3\u03c730\u222b \u221e0U1QII\u03c1 d\u03c1+1f3\u03c730\u222b \u221e0U1QI\u03c1 d\u03c1\u2212 S1f1\u2212 f) ,= \u2212(1\u2212 f)bf2S1S20\u2212S0b2\u222b \u221e0U1QII\u03c1 d\u03c1+S0(1\u2212 f)b2\u222b \u221e0U1QI\u03c1 d\u03c1 .In conclusion, we have constructed a two-term asymptotic spot solution tothe core problem (3.3) in the limit S \u2192 0. We summarize our result asfollows:Principal Result 1. For S \u223c \u03bd12 (S0 + \u03bdS1 + ...), where \u03bd \u2261 \u22121ln \u000f , the coreproblem (3.3) has an asymptotic solution in the formU \u223c \u03bd12 (U0+\u03bdU1+\u00b7 \u00b7 \u00b7 ), V \u223c \u03bd\u2212 12 (V0+\u03bdV1+\u00b7 \u00b7 \u00b7 ), \u03c7 \u223c \u03bd\u2212 12 (\u03c70+\u03bd\u03c71+\u00b7 \u00b7 \u00b7 ) ,(3.29)where U0(\u03c1), U1(\u03c1), V0(\u03c1) and V1(\u03c1) are defined by:U0 =\u03c9f\u03c70, U1 = \u2212\u03c71f\u03c720\u03c9 \u22121f3\u03c730((1\u2212 f)U1QII \u2212 U1QI) ,V0 = \u03c70 , V1 = \u03c71 +1f2\u03c70((1\u2212 f)V1Q \u2212 \u03c9) .253.2. Linear Stability AnalysisHere U1QI , U1QII , and V1Q are the unique radially symmetric solutions to(3.26), (3.27), and (3.24), respectively. In addition, \u03c9 is the unique positiveradially symmetric solution to (3.17), while \u03c70 and \u03c71 are defined by\u03c70 =(1\u2212 f)f2bS0,\u03c71 = \u2212(1\u2212 f)bf2S1S20\u2212S0b2\u222b \u221e0U1QII\u03c1 d\u03c1+S0(1\u2212 f)b2\u222b \u221e0U1QI\u03c1 d\u03c1 .3.2 Linear Stability AnalysisIn this section, we perform a linear stability analysis for the equilibrium spotsolution that we have just constructed. We perturb the steady state solu-tions and derive the singularly perturbed eigenvalue problem. By analyzingthis eigenvalue problem we get the stability threshold. As we mentioned inSection 2.3, since the steady state solutions we have just derived is periodicwith respect to the lattice \u039b, solving the perturbed system is equivalentto finding eigenfunctions of the operator \u2206 + f(x), where f(x) is a 2 \u00d7 2periodic matrix with respect to the lattice \u039b. As a result of the general-ized Bloch theory, instead of solving the perturbed system on R2, we needonly consider the eigenvalue problem within the primitive cell \u2126 with theBloch boundary conditions (2.15), which involves a Bloch vector in the FirstBrillouin Zone.We begin perturbing the equilibrium solutions asu = ue + e\u03bbt\u03c6 , v = ve + e\u03bbt\u03b7 . (3.30)We linearize the equations around this steady state solution and obtain, toleading order, the singularly perturbed eigenvalue problem\u03bb(\u03c6\u03c4\u03b7)=(\u000f2\u2206\u03c6D\u2206\u03b7)+(\u22121 + 2fueve fu2e\u000f\u22122 \u2212 2\u000f\u22122ueve \u000f\u22122u2e)(\u03c6\u03b7), (3.31)263.2. Linear Stability Analysisfor x \u2208 \u2126, where \u03c6 and \u03b7 satisfy the Bloch boundary conditions (2.15) forsome k in the first Brillouin Zone.Since the equilibrium solution is localized then, as similar to the con-struction of the equilibrium spot solution, we introduce an inner region nearthe core of the spot. As such, we introduce the following inner coordinatesand scale the system asy = \u000f\u22121x , \u03a6(y) =\u03c6(x)D=\u03c6(\u000fy)D, N(y) = \u03b7(x) = \u03b7(\u000fy) . (3.32)We still look for radially symmetric solutions \u03a6(y) = \u03a6(\u03c1), N(y) = N(\u03c1),where \u03c1 = |y|. Then, to leading order, the inner system becomes\u2206\u03c1\u03a6\u2212 (\u03bb+ 1)\u03a6 + 2fUV \u03a6 + fU2N = 0 , \u03a6\u2032(0) = 0 ; \u03a6\u2192 0 as \u03c1\u2192\u221e ,(3.33)\u2206\u03c1N + (1\u2212 2UV )\u03a6\u2212 U2N = 0 , N \u2032(0) = 0 ; N \u2192 C ln |y|+B(C, f) ,(3.34)as \u03c1 \u2192 \u221e. We remark here that since this is a linear system for a fixed f ,it follows that the ratio BC is a constant. In addition, we remark that theasymptotic boundary condition for \u03a6 is appropriate provided that Re(\u03bb +1) > 0, i.e. that \u03bb is not in the continuous spectrum. Finally, we obtain thefollowing identity by integrating the equation for N to getC =\u222b \u221e0(U2N \u2212 (1\u2212 2UV )\u03a6)\u03c1 d\u03c1 . (3.35)In terms of C, in the outer region the second equation of (3.31) is ap-proximated as\u2206\u03b7 \u2212\u03c4\u03bbD\u03b7 = 2piC\u03b4(x) , (3.36)with the Bloch boundary conditions (2.15) for some Bloch vector k. In termsof the reduced-wave Bloch Green function (2.16), we can write \u03b7 as\u03b7 = \u22122piCG\u00b5,k(x) , \u00b5 =\u221a\u03c4\u03bbD\u223c O(\u03bd12 ) . (3.37)273.2. Linear Stability AnalysisTherefore, as x\u2192 0, \u03b7(x) has the asymptotic behaviour\u03b7(x) \u223c \u22122piC(\u221212piln |x|+R\u00b5,k) \u223c C ln |x| \u2212 2piCR\u00b5,k . (3.38)We know from Lemma 3 that if |k| is bounded away from the origin, thenwe have R\u00b5,k = R0,k +O(\u00b52) = R0,k +O(\u03bd).Next, we asymptotically match (3.38) with the far-field behavior of theinner solution (3.34), which is given by\u03b7 \u223c C ln |x|+C\u03bd+B . (3.39)Upon comparing (3.38) and (3.39), we conclude thatC(1 + 2pi\u03bdR0,k +O(\u03bd2)) = \u2212\u03bdB . (3.40)Next, we will calculate the asymptotic solution to (3.33) and (3.34) for\u03bd small. Since we have UV \u223c O(1), U2 \u223c O(\u03bd) and C \u223c \u03bdB, this suggeststhat we expand the solutions as\u03a6 \u223c \u03bd(\u03a60 + \u03bd\u03a61 + \u00b7 \u00b7 \u00b7 ) , N \u223c N0 + \u03bdN1 + \u00b7 \u00b7 \u00b7 ,B \u223c B0 + \u03bdB1 + \u00b7 \u00b7 \u00b7 , C \u223c \u03bd(C0 + \u03bdC1 + \u00b7 \u00b7 \u00b7 ) .Upon substituting these expansions into (3.33) and (3.34), we obtain toleading order that\u2206\u03a60 \u2212 (\u03bb+ 1)\u03a60 + 2fU0V0\u03a60 + fU20N0 = 0 , \u2206N0 = 0 ,\u03a6\u20320(0) = N\u20320(0) = 0 ; \u03a60 \u2192 0 , N0 \u2192 B0 as |y| \u2192 \u221e .Then, upon substituting the expressions for U0 and V0, as given in PrincipalResult 1, we getL0\u03a60 = \u03bb\u03a60 \u2212B0f\u03c720\u03c92 , N0 = B0 . (3.41)283.2. Linear Stability AnalysisFrom (3.40), we conclude thatB0 = \u2212C0 = N0 , B1 = \u2212C1 \u2212 2piR0,kC0 . (3.42)At next order, we have that \u03a61 satisfies\u2206\u03a61 \u2212 (\u03bb+ 1)\u03a61 + 2f(U0V1 + U1V0)\u03a60 + 2fU0V0\u03a61+ fU20N1 + 2fU0U1N0 = 0 ; \u03a6\u20321(0) = 0 , \u03a61 \u2192 0 as |y| \u2192 \u221e ,(3.43)while N1 satisfies\u2206N1 + (1\u2212 2U0V0)\u03a60 \u2212 U20N0 = 0 ,N \u20321(0) = 0 , N1 \u2192 C0 ln |y|+B1 as |y| \u2192 \u221e .Upon substituting the results for U0 and V0 from Principal Result 1, weobtain from the divergence theorem on the N1 equation thatC0 =(1 +bf2\u03c720)\u22121 \u222b \u221e0(2f\u03c9 \u2212 1)\u03a60\u03c1 d\u03c1 . (3.44)With C0 now determined, we substitute it back into the equation for \u03a60 toobtainL0\u03a60 = \u03bb\u03a60 +fb+ f2\u03c720(2f\u222b \u221e0\u03c9\u03a60\u03c1 d\u03c1\u2212\u222b \u221e0\u03a60\u03c1 d\u03c1)\u03c92 . (3.45)In contrast to the nonlocal eigenvalue problems (NLEP\u2019s) analyzed in[8], this NLEP is more intricate as it involves two nonlocal terms. In order toobtain an NLEP with only one nonlocal term, we integrate (3.45) to derive\u222bR2L0\u03a60dx = 0\u2212 2pi\u222b \u221e0\u03a60\u03c1 d\u03c1+ 4pi\u222b \u221e0\u03c9\u03a60\u03c1 d\u03c1= 2pi\u03bb\u222b \u221e0\u03a60\u03c1 d\u03c1+2pibfb+ f2\u03c720(2f\u222b \u221e0\u03c9\u03a60\u03c1 d\u03c1\u2212\u222b \u221e0\u03a60\u03c1 d\u03c1),293.2. Linear Stability Analysiswhich leads to(\u03bb+ 1\u2212bfb+ f2\u03c720)\u222b \u221e0\u03a60\u03c1 d\u03c1 = (2\u22122bb+ f2\u03c720)\u222b \u221e0\u03a60\u03c9\u03c1 d\u03c1 . (3.46)We substitute this back into (3.45) to getL0\u03a60 \u22122b(\u03bb+ 1\u2212 f)(\u03bb+ 1)(b+ f2\u03c720)\u2212 bf\u222b\u221e0 \u03a60\u03c9\u03c1 d\u03c1\u222b\u221e0 \u03c92\u03c1 d\u03c1\u03c92= L0\u03a60 \u22122(\u03bb+ 1\u2212 f)(\u03bb+ 1)(1 + (1\u2212ff )2 bS20)\u2212 f\u222b\u221e0 \u03a60\u03c9\u03c1 d\u03c1\u222b\u221e0 \u03c92\u03c1 d\u03c1\u03c92 = \u03bb\u03a60 .To analyze this NLEP we use some previous results on NLEP\u2019s similarto the one above, which can be found in [16]. From this previous theory,we know the stability threshold occurs when \u03b2(0) = 1, where \u03b2(\u03bb) is themultiplier of the nonlocal term defined by\u03b2(\u03bb) =2(\u03bb+ 1\u2212 f)(\u03bb+ 1)(1 + (1\u2212ff )2 bS20)\u2212 f.This condition determines a critical value Sc0 of S0 as S2c0 =(1\u2212f)bf2 . Then,by (3.8), this determines the following leading order result for the stabilitythreshold:Dc =Dc0\u03bd+Dc1 + \u00b7 \u00b7 \u00b7 , Dc0 =f2(1\u2212 f)bE2|\u2126|24pi2. (3.47)We remark that at this leading-order stability threshold we have \u03a60 = \u03c9,which then yields f2\u03c720 = (1\u2212 f)b and C0 = f\u03c720 =1\u2212ff b = \u2212B0.It is critical here to emphasize that the leading order stability thresholdis independent of the details of the lattice \u039b, and does not depend on theBloch vector k. In order to determine the effect of the lattice on the stabilitythreshold we must proceed to one higher order.We now continue the calculation to one higher order. At the leadingorder stability threshold we can simplify the equation for \u03a6 and N , and weexpand the eigenvalue as \u03bb = \u03bd\u03bb1 + \u00b7 \u00b7 \u00b7 in order to calculate the spectrum303.2. Linear Stability Analysisnear the origin. At the leading order threshold, the equation for N1 is now\u2206N1 =1f\u03c92 \u2212 \u03c9 , N \u20321(0) = 0 , N1 \u2192 C0 ln |y|+B1 as |y| \u2192 \u221e .(3.48)Upon recalling the definition of V1P , it follows thatN1 =V1Pf+B1 . (3.49)Since we are seeking the second order term then, as similar to the construc-tion of the equilibrium spot solution, we will need to analyze the third orderequation and impose a solvability condition. At the leading order stabilitythreshold, the equation for the third-order term N2 becomes\u2206N2 + (1\u22122f\u03c9)\u03a61 \u2212 3\u03c92V1Pf3\u03c720\u2212B1f2\u03c720\u03c92 \u2212 2\u03c71f\u03c70\u03c92 = 0 ,N \u20322(0) = 0 , N2 \u2192 C1 ln |y|+B2 , as |y| \u2192 \u221e .The solvability condition for this equation yields thatC1 = \u2212\u222b \u221e0(1\u22122f\u03c9)\u03a61\u03c1 d\u03c1+3f3\u03c720\u222b \u221e0\u03c92V1P\u03c1 d\u03c1+B11\u2212 f+2\u221ab1\u2212 f\u03c71 .(3.50)Upon combining this equation with \u22122piR0,kC0 \u2212 C1 = B1, we derive thatB1 =(1\u2212 f2\u2212 f)(\u222b \u221e0(1\u22122f\u03c9)\u03a61\u03c1 d\u03c1\u22123f3\u03c720\u222b \u221e0\u03c92V1P\u03c1 d\u03c1+2f3\u03c720\u222b \u221e0U1P\u03c1 d\u03c1\u2212 2piR0,k1\u2212 ffb+ 2\u221ab1\u2212 fS1) .(3.51)Once again, we want to eliminate the term\u222b\u221e0 \u03a61\u03c1 d\u03c1. This can be doneby using the same method as done previously above. To do so, we substitute(3.51) into the equation of \u03a61 to getL0\u03a61 \u2212 \u03bb1\u03c9 +3f2\u03c720\u03c92V1P +2\u03c71\u03c70\u03c92 +B1f\u03c720\u03c92 = 0 . (3.52)313.2. Linear Stability AnalysisWe then integrate both sides of this expression we obtain\u222b \u221e0\u03a61\u03c1 d\u03c1 =\u222b \u221e0\u03a61\u03c9\u03c1 d\u03c1\u22122\u2212 f2\u2212 2f\u03bb1b+32f2\u03c720\u222b \u221e0\u03c92V1P\u03c1 d\u03c1+b\u03c71\u03c70\u2212 pibR0,k .(3.53)Then we substitute (3.53) and (3.51) back into (3.52) to concludeL\u03a61 \u2261 L0\u03a61 \u2212\u222b\u221e0 \u03a61\u03c9\u03c1 d\u03c1\u222b\u221e0 \u03c92\u03c1 d\u03c1\u03c92 = \u03bb1(\u03c9 +f2\u2212 2f\u03c92)\u22123f2\u03c720\u03c92V1P+(32f2\u03c720b\u222b \u221e0\u03c92V1P\u03c1 d\u03c1\u2212\u03c71\u03c70+ piR0,k)\u03c92 .(3.54)Finally, \u03bb1 is determined by imposing a solvability condition on \u03a61 in (3.54).The adjoint operator of L is simplyL?\u03a8 \u2261 L0\u03a8\u2212 \u03c9\u222b\u221e0 \u03c92\u03a8\u03c1 d\u03c1\u222b\u221e0 \u03c92\u03c1 d\u03c1. (3.55)It is readily verified that if we define \u03a8? = w+\u03c1w\u2032\/2, then we have L?\u03a8\u2217 = 0.The null space of L? was first identified in [8]. By imposing the Fredholmalternative on (3.54) we get\u03bb1\u222b \u221e0(\u03c9 +f2\u2212 2f\u03c92)\u03a8\u2217\u03c1 d\u03c1\u221232f2\u03c720\u222b \u221e0\u03c92V1P\u03a8\u2217\u03c1 d\u03c1 (3.56)+ (32f2\u03c720b\u222b \u221e0\u03c92V1P\u03c1 d\u03c1\u2212\u03c71\u03c70+ piR0,k)\u222b \u221e0\u03c92\u03a8\u2217\u03c1 d\u03c1 = 0 .This expression can be simplified considerably by using the following iden-tities:\u222b \u221e0\u03c92\u03a8\u2217\u03c1 d\u03c1 =\u222b \u221e0(L0\u03c9)(L\u221210 \u03c9)=\u222b \u221e0\u03c92\u03c1 d\u03c1 = b ,\u222b \u221e0\u03c9\u03a8\u2217\u03c1 d\u03c1 =\u222b \u221e0\u03c1\u03c9(\u03c9 +\u03c12\u03c9\u2032) d\u03c1 =\u222b \u221e0\u03c92\u03c1 d\u03c1+14\u222b \u221e0[\u03c92]\u2032\u03c12 d\u03c1 =b2,\u222b \u221e0\u03c92V1P\u03a8\u2217\u03c1 d\u03c1 =\u222b \u221e0(L0U1P )(L\u221210 \u03c9)=\u222b \u221e0U1P\u03c9\u03c1 d\u03c1 ,323.2. Linear Stability Analysis2\u222b \u221e0\u03c9U1P\u03a8\u2217\u03c1 d\u03c1\u2212\u222b \u221e0U1P\u03c1 d\u03c1 =\u222b \u221e0\u03c92V1P\u03c1 d\u03c1 .In this way, we determine the spectrum near the origin in the spectralplane as\u03bb1 = 2(1\u2212 f)(\u03c71\u03c70\u2212 piR0,k +32f2\u03c720b\u222b \u221e0U1P\u03c1 d\u03c1). (3.57)Since we are seeking a two-term expansion for the stability threshold Dc ofD, we need to write \u03c70 and \u03c71 in term of D. In Principal Result 1, \u03c7 waswritten in term of S, and S is in term of D asS = \u03bd12 (S0 + \u03bdS1 + \u00b7 \u00b7 \u00b7 ) =E|\u2126|2pi\u221aD=E|\u2126|2pi[D0(1 + \u03bdD1D0+ \u00b7 \u00b7 \u00b7)]\u2212 12= \u03bd\u221212E|\u2126|D\u2212 1202pi(1\u2212\u03bd2D1D0+ \u00b7 \u00b7 \u00b7).Now with Dc0 =f2(1\u2212f)bE2|\u2126|24pi2 , we have Sc1 = \u221212S0Dc1Dc0, so that\u03c71\u03c70=f\u03c70b(\u22121f3\u03c730\u222b \u221e0U1P\u03c1 d\u03c1\u2212 S1f1\u2212 f).Therefore, we have\u03bb1 = 2(1\u2212 f)(12(1\u2212 f)b2\u222b \u221e0U1P\u03c1 d\u03c1+2pi2(1\u2212 f)bf2E2|\u2126|2D1 \u2212 piR0,k).(3.58)Notice that this is a continuous band of spectra parametrized by theBloch vector k. This is illustrated schematically in Figure 3.2. As proved inLemma 1 and Lemma 2, R0,k is real and tends to infinity as |k| \u2192 0. Thisshows from (3.58) that \u03bb1 is real, and leaves the ball of radius O(\u03bd) \u001c 1near the origin along the negative real axis as |k| \u2192 0.Therefore, in order to have stability, we need \u03bb1 < 0, \u2200k \u2208 \u2126\u2217. Wesummarize our result as follows:Principal Result 2. In the limit \u000f \u2192 0, D \u223c O( 1\u03bd ), we have constructed333.2. Linear Stability AnalysisFigure 3.1: The Continuous Band of Spectraperiodic spot solutions(with respect to a fixed Bravais lattice \u039b) in PrincipalResult 1. The linearized operator around this solution has a continuousspectrum within an O(\u03bd) neighbourhood of the origin and is parametrized bya Bloch vector k \u2208 \u2126\u2217\\0:\u03bb(k) = 2\u03bd(1\u2212 f)(12(1\u2212 f)b2\u222b \u221e0U1P\u03c1 d\u03c1+2pi2(1\u2212 f)bf2E2|\u2126|2D1 \u2212 piR0,k).(3.59)To have linear stability, we need \u03bb(k) < 0, \u2200k \u2208 \u2126\u2217\\0, which gives a twoterm asymptotic expansion for the stability threshold Dc:Dc =Dc0\u03bd+Dc1 + \u00b7 \u00b7 \u00b7 , where Dc0 =f2E2|\u2126|24pi2(1\u2212 f)b, andDc1 = mink\u2208\u2126\u2217\\0{f2E2|\u2126|22pi2(1\u2212 f)b(piR0,k \u221212(1\u2212 f)b2\u222b \u221e0U1P\u03c1 d\u03c1)} ,343.3. A Quick Derivation of the Stability Threshold= mink\u2208\u2126\u2217\\0{Dc0(2piR0,k \u22121(1\u2212 f)b2\u222b \u221e0U1P\u03c1 d\u03c1)}= mink\u2208\u2126\u2217\\0{Dc0(2piR0,k +1(1\u2212 f)b2\u222b \u221e0U1QI\u03c1 d\u03c1\u22121b2\u222b \u221e0U1QII\u03c1 d\u03c1)} .We remark here that since R0,k is real-valued, then so is the thresholdDc,which is what we should expect. As shown above, the second order term inDc depends on the lattice \u039b. In order to compare the stability threshold ondifferent lattices, we will fix |\u2126| = 1. This leads to the following optimalitycriterion:Principal Result 3. With fixed primitive cell of area unity, the optimallattice arrangement \u039bop is the one with the largest stability threshold:\u039bop = arg max\u039b, |\u2126|=1{Dc(\u039b)} = arg max\u039b, |\u2126|=1{ mink\u2208\u2126\u2217\\0{R0,k}} . (3.60)Some numerical results to identify the optimal lattice arrangement isgiven in Section 5.2.3.3 A Quick Derivation of the StabilityThresholdIn this section, we give another much more expedient way to derive thestability threshold. Recall that in the inner core problem (3.3), S is a pa-rameter in the asymptotic boundary condition. More specifically, if we viewS as a parameter of the solution, then\u2206\u03c1U(|y|, S)\u2212 U(|y|, S) + fU2(|y|, S)V (|y|, S) = 0 ,\u2202U\u2202\u03c1(0, S) = 0 , U \u2192 0 as |y| \u2192 \u221e ,\u2206\u03c1V (|y|, S) + U(|y|, S)\u2212 U2(|y|, S)V (|y|, S) = 0 ,\u2202V\u2202\u03c1(0, S) = 0 , V \u2192 S ln |y|+ \u03c7(S, f) as |y| \u2192 \u221e .353.3. A Quick Derivation of the Stability ThresholdUpon taking the partial derivative of U and V with respect to S, we get\u2206(\u2202U\u2202S\u2202V\u2202S)+(\u22121 + 2fUV fU21\u2212 2UV \u2212U2)(\u2202U\u2202S\u2202V\u2202S)=(00), (3.61)with the boundary conditions\u22022U\u2202S\u2202\u03c1(0, S) = 0 ,\u2202U\u2202S\u2192 0 , as |y| \u2192 \u221e ,\u22022V\u2202S\u2202\u03c1(0, S) = 0 ,\u2202V\u2202S\u2192 ln |y|+\u2202\u03c7\u2202S(S, f) , as |y| \u2192 \u221e .Then, we observe that \u2202U\u2202S and\u2202V\u2202S are, up to a scalar multiple, thesolution to the perturbed core problem (3.33) and (3.34) when \u03bb = 0. Tofix the boundary conditions in (3.33) and (3.34) for N(y), we must chooseS appropriately. This constraint on S to hold when \u03bb = 0 is the stabilitythreshold we are seeking. Since the solution to (3.33) and (3.34) is unique upto a constant scaling, then upon comparing the boundary conditions with(3.33) and (3.34), it follows that the stability threshold Sc occurs whenBC=\u2202\u03c7\u2202S(Sc, f) . (3.62)Then, together with (3.40), we have\u2202\u2202S\u03c7(Sc, f) = \u22121\u03bd\u2212 2piR0,k +O(\u03bd) . (3.63)We then use the expansion for \u03c7 from Principal Result 1, which we write as\u03c7(S, f) = \u03bd\u221212(b(1\u2212 f)f21S0+ \u03bd(\u2212(1\u2212 f)bf2S1S20\u2212S0(1\u2212 f)b2\u222b \u221e0U1P\u03c1 d\u03c1) + \u00b7 \u00b7 \u00b7),S = \u03bd12 (S0 + \u03bdS1 + \u00b7 \u00b7 \u00b7 ) ,to derive\u03c7(S, f) =b(1\u2212 f)f21S\u2212\u222b\u221e0 U1P\u03c1 d\u03c1(1\u2212 f)b2S +O(\u03bd) , (3.64)363.3. A Quick Derivation of the Stability Threshold=b(1\u2212 f)f21S+(1b2(1\u2212 f)\u222b \u221e0U1QI\u03c1 d\u03c1\u22121b2\u222b \u221e0U1QII\u03c1 d\u03c1)S +O(\u03bd) .Upon taking the partial derivative we obtain\u2202\u2202S\u03c7(S, f) = \u2212b(1\u2212 f)f21S2\u2212\u222b\u221e0 U1P\u03c1 d\u03c1(1\u2212 f)b2+O(\u03bd) , (3.65)= \u22121\u03bdb(1\u2212 f)f21S20(1\u2212 2S1S0\u03bd + \u00b7 \u00b7 \u00b7 )\u2212\u222b\u221e0 U1P\u03c1 d\u03c1(1\u2212 f)b2+O(\u03bd) ,= \u22121\u03bdb(1\u2212 f)f21S20+ (2b(1\u2212 f)f2S1S30\u2212\u222b\u221e0 U1P\u03c1 d\u03c1(1\u2212 f)b2) +O(\u03bd) .Upon comparing this expression with (3.63), we obtain from equating powersof \u03bd that\u22121 = \u2212b(1\u2212 f)f21S2c0,\u22122piR0,k = 2b(1\u2212 f)f2Sc1S3c0\u2212\u222b\u221e0 U1P\u03c1 d\u03c1(1\u2212 f)b2,= 2b(1\u2212 f)f2Sc1S3c0+1b2(1\u2212 f)\u222b \u221e0U1QI\u03c1 d\u03c1\u22121b2\u222b \u221e0U1QII\u03c1 d\u03c1 .In this way, we can solve for Sc0 and Sc1, then obtain the corrections Dc0and Dc1 to the stability threshold from expanding the relation (3.8) betweenS and D. This yields the same result for the stability threshold as derivedin Principal Result 2. We remark that although this simple derivation isable to quickly isolate the stability threshold, it is unable to give any preciseaccount of the nature of the spectrum of the linearized operator near theorigin when D is near the leading-order stability threshold.37Chapter 4Spot Patterns for theBrusselator on a FiniteDomainIn this chapter, we first construct N -spot solutions to the Brusselator model,formulated asut = \u000f2\u2206u\u2212 u+ f(u2v + 2\u000f2Euv + \u000f4Ev) ,\u03c4vt = D\u2206v + E + \u000f\u22122(u\u2212 u2v)\u2212 2Euv \u2212 \u000f2E2v ,(4.1)on a finite domain x \u2208 \u2126 \u2282 R2, with no-flux boundary conditions\u2202nu(x) = \u2202nv(x) = 0 , x \u2208 \u2202\u2126 , (4.2)where n is the outer normal vector on \u2202\u2126. After constructing such multi-spot patterns, we then perform a linear stability analysis to calculate astability threshold corresponding to a zero-eigenvalue crossing.The asymptotic construction of the N -spot pattern and the linear sta-bility analysis is very similar to that for the periodic case. One of the keydifferences between the periodic and finite-domain problems, is that for theperiodic case we need only construct a one-spot solution in one primitivecell. In contrast, for the finite domain case, spots interact with each otherin the domain through the Green\u2019s function. As a result, a Neumann Greenmatrix together with its eigenvalues and eigenvectors play a key role in theanalysis.384.1. The N -Spot Solutions4.1 The N-Spot SolutionsIn this section, our goal is to construct N -spot solutions where u(x) concen-trates around N distinct O(\u000f) balls centred at N given points x1,x2, ...,xn \u2208\u2126. The position of these N points are not arbitrary, but satisfy some con-ditions to be derived and explained below.We first introduce local coordinates around each of these N points inthe formy = \u000f\u22121(x\u2212 xj) , Uj(y) =u(x)\u221aD, Vj(y) =\u221aDv(x) , j = 1, 2, ..., N .(4.3)Then, in an inner region around xj , we look for a radially symmetric solutionin the form Uj(y) = Uj(\u03c1), Vj(y) = Vj(\u03c1), where \u03c1 = |y|. To leading order,we get the core problem\u2206\u03c1Uj \u2212 Uj + fU2j Vj = 0 , \u2206\u03c1Vj + Uj \u2212 U2j Vj = 0 , 0 < \u03c1 <\u221e , (4.4)with the boundary conditionsU \u2032j(0) = V\u2032j (0) = 0 , Uj \u2192 0 , Vj \u2192 Sj ln \u03c1+ \u03c7(Sj , f) , as \u03c1\u2192\u221e .(4.5)Here the introduction of the source strength Sj and the function \u03c7(Sj , f)is the same as in (3.4). Upon integrating \u2206Vj and using the divergencetheorem we get2piSj =\u222bR2(U2j Vj \u2212 Uj) dx =\u222b 2pi0d\u03b8\u222b \u221e0(U2j Vj \u2212 Uj)\u03c1 d\u03c1 ,which yieldsSj =\u222b \u221e0(U2j Vj \u2212 Uj)\u03c1 d\u03c1 . (4.6)In the outer region, given that u is localized around {xi}Ni=1, then byusing the relation (4.6) we obtain that the leading order outer solution v394.1. The N -Spot SolutionssatisfiesD\u2206v + E = 2pi\u221aDN\u2211i=1Si\u03b4(x\u2212 xi) , x \u2208 \u2126 , (4.7)with \u2202nv = 0 for x \u2208 \u2202\u2126. Upon integrating (4.7) over \u2126, and by using theno-flux boundary conditions (4.2), we get the solvability conditionE =2pi\u221aD|\u2126|N\u2211j=1Sj . (4.8)To represent the solution v, it is convenient to introduce the Neumann Greenfunction G(x, \u03be), which satisfies\u2206xG0(x, \u03be) =1|\u2126|\u2212 \u03b4(x\u2212 \u03be) , (4.9)\u2207xG0(x, \u03be) \u00b7 n = 0 , \u2200x \u2208 \u2202\u2126 ,\u222b\u2126G0(x, \u03be) dx = 0 , (4.10)G0(x, \u03be)\u2192 \u221212piln |x\u2212 \u03be|+R0(\u03be) , as x\u2192 \u03be , (4.11)where n is the outer normal vector to \u2202\u2126. We remark here that R(\u03be) is theregular part of the Green function while \u2212 12pi ln |x\u2212 \u03be| is the singular part.The right hand side of the equation 1|\u2126| \u2212 \u03b4(x \u2212 \u03be) is consistent with theno-flux boundary conditions \u2207xG0(x, \u03be) \u00b7n = 0, \u2200x \u2208 \u2202\u2126. We also imposethe uniqueness condition,\u222b\u2126G0(x, \u03be) dx = 0, since the solution to the PDEwith the no-flux boundary conditions is only unique up to a constant.In terms of G0, the solution v to (4.7) can be represented asv(x) = \u2212N\u2211i=12piSi\u221aDG0(x,xi) + c , (4.12)where c is some constant. Then, by letting x\u2192 xj , and by matching to theinner solution near each spot, we obtain the following nonlinear system of404.1. The N -Spot SolutionsN equations for Sj , j = 1, . . . , N , and for c:Sj\u03bd+ \u03c7(Sj) = \u22122piR0(xj)Sj \u2212 2pi\u2211i 6=jG0(xi,xj)Si +\u221aDc , 1 \u2264 j \u2264 N .(4.13)The remaining equation to complete the system is (4.8). It is convenient torewrite this system in matrix form by introducing the following notation:S \u2261\uf8eb\uf8ec\uf8ec\uf8edS1...SN\uf8f6\uf8f7\uf8f7\uf8f8 , e \u2261\uf8eb\uf8ec\uf8ec\uf8ed1...1\uf8f6\uf8f7\uf8f7\uf8f8 , \u03c7(S, f) \u2261\uf8eb\uf8ec\uf8ec\uf8ed\u03c7(S1, f)...\u03c7(SN , f)\uf8f6\uf8f7\uf8f7\uf8f8 ,G \u2261\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8edR0(x1) G0(x1,x2) \u00b7 \u00b7 \u00b7 G0(x1,xN )G0(x2,x1) R0(x2) \u00b7 \u00b7 \u00b7.... . ....G0(xN ,x1) \u00b7 \u00b7 \u00b7 R0(xN )\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8.We shall refer to G as the Neumann Green matrix of x1,x2, ...,xN . Then,the N matching conditions (4.13) and the solvability condition (4.8) can bewritten in matrix form asS + \u03bd\u03c7(S, f) = \u2212\u03bd2piGS + \u03bd\u221aDce ,eTS =|\u2126|E2pi\u221aD.(4.14)For simplicity we will assume that the N spots have a common sourcestrength S, i.e. for Sj = S, j = 1, 2, ..., N . For such a pattern, we have thatS = Se, \u03c7(S, f) = \u03c7(S, f)e, and that (4.14) reduces toGS = SGe = \u221212pi\u03bd(S + \u03bd\u03c7(S, f)\u2212 \u03bd\u221aDc)e , S =E|\u2126|2piN\u221aD. (4.15)Therefore, it follows that for such a common source strength pattern toexist we must have that e is an eigenvector of G(x1,x2, ...,xN ). We remarkthat the existence of such a special eigenvalue does not generally occur fora pattern of N arbitrarily-located spots. Since the Green function satisfies414.1. The N -Spot Solutionsthe usual reciprocity condition, it follows that G is a symmetric matrix andcan be diagonalized with an orthonormal basis as 1\u221aNe, {qj}Nj=2, i.e.G(1\u221aNe) = k1(1\u221aNe) , Gqj = kjqj , j = 2, 3, ..., N, (4.16)qTj1\u221aNe = 0 , j = 2, 3, ..., N , qTj qi = 0 , \u2200 2 \u2264 i, j \u2264 N .Upon using this relation, we obtain for a common source strength patternthat (4.14) reduces to12pi\u03bd(S + \u03bd\u03c7(S, f)\u2212 \u03bd\u221aDc) = \u2212Sk1 . (4.17)Next, since the stability threshold occurs in the regime where D \u223cO(\u03bd\u22121), it follows that S \u223c O(\u03bd12 ) at a zero eigenvalue crossing. By fol-lowing the same procedure as for the periodic problem, we can calculate thesmall S asymptotics of the solution to the core problem (4.4) as follows:Principal Result 4. For S \u223c \u03bd12 (S0 + \u03bdS1 + \u00b7 \u00b7 \u00b7 ), where \u03bd \u2261 \u2212 1ln \u000f , theradially symmetric asymptotic solutions to the core problem (4.4) in an O(\u000f)ball centred at xj is given by:Uj \u223c \u03bd12 (Uj0 + \u03bdUj1 + \u00b7 \u00b7 \u00b7 ) , Vj \u223c \u03bd\u2212 12 (Vj0 + \u03bdVj1 + \u00b7 \u00b7 \u00b7 ) ,\u03c7 \u223c \u03bd\u221212 (\u03c70 + \u03bd\u03c71 + \u00b7 \u00b7 \u00b7 ) , (4.18)where Uj0(\u03c1), Uj1(\u03c1), Vj0(\u03c1) and Vj1(\u03c1) are defined byUj0 =\u03c9f\u03c70, Uj1 = \u2212\u03c71f\u03c720\u03c9 \u22121f3\u03c730((1\u2212 f)U1QII \u2212 U1QI) , (4.19)Vj0 = \u03c70 , Vj1 = \u03c71 +1f2\u03c70((1\u2212 f)V1Q \u2212 \u03c9) . (4.20)Here U1QI , U1QII , V1Q are the unique radially symmetric solutions to (3.26),(3.26), (3.24) as before, \u03c9 is the unique radially symmetric solution to424.2. Linear Stability Analysis(3.17), while \u03c70 and \u03c71 are defined as\u03c70 =(1\u2212 f)f2bS0, (4.21)\u03c71 = \u2212(1\u2212 f)bf2S1S20\u2212S0b2\u222b \u221e0U1QII\u03c1 d\u03c1+S0(1\u2212 f)b2\u222b \u221e0U1QI\u03c1 d\u03c1 . (4.22)We remark here that since the source strength is the same for each j,then so are the inner solutions. Moreover, we remark that such patterns caneither be true steady-state solutions if the spot locations satisfy some furthercondition, or simply quasi-steady patterns that can persist for very long timeintervals provided that there is no unstable O(1) eigenvalue in the spectrumof the linearization. The analysis of the spectrum of the linearization isanalyzed in the next section. This completes our construction of an N -spotsolution where the spots have a common source strength.4.2 Linear Stability AnalysisWe denote the N -spot solution constructed above as ue(x), ve(x) and weintroduce the perturbationu(x) = ue(x) + e\u03bbt\u03c6 , v(x) = ve(x) + e\u03bbt\u03b7 . (4.23)Upon substituting (4.23) into (4.1), we linearize around the N -spot solutionto obtain the singularly perturbed problem (3.31), written again as\u03bb(\u03c6\u03c4\u03b7)=(\u000f2\u2206\u03c6D\u2206\u03b7)+(\u22121 + 2fueve fu2e\u000f\u22122 \u2212 2\u000f\u22122ueve \u000f\u22122u2e)(\u03c6\u03b7).In the inner region around each xj , we introduce the local variables asy = \u000f\u22121(x\u2212 xj) , \u03a6j(y) =\u03c6(\u000fy + xj)D, Nj(y) = \u03b7(\u000fy + xj) . (4.24)We look for radially symmetric solution of the form \u03a6j(y) = \u03a6j(\u03c1), Nj(y) =434.2. Linear Stability AnalysisNj(\u03c1), where \u03c1 = |y|. Then, the inner problem near the j-th spot at xjreduces asymptotically to\u2206\u03c1\u03a6j \u2212 (\u03bb+ 1)\u03a6j + 2fUjVj\u03a6 + fU2jNj = 0 , (4.25)\u2206\u03c1Nj + (1\u2212 2UjVj)\u03a6j \u2212 U2jNj = 0 ,\u03a6\u2032j(0) = N\u2032j(0) = 0 ; \u03a6j \u2192 0 , Nj \u2192 Cj ln |y|+Bj(Cj , f) , as \u03c1\u2192\u221e .We remark here that for a fixed f , the ratio BjCj is a constant since the systemis linear. The solvability condition for the Nj equation yieldsCj =\u222b \u221e0(U2jNj \u2212 (1\u2212 2UjVj)\u03a6j)\u03c1 d\u03c1 . (4.26)Since both ue(x) and \u03c6(x) are localized near {xj}Nj=1, the outer approx-imation for the eigenfunction component \u03b7(x) satisfies the leading orderproblem\u2206\u03b7 \u2212\u03c4\u03bbD\u03b7 = 2piN\u2211i=1Ci\u03b4(x\u2212 xi) . (4.27)Notice that when \u03bb = 0, corresponding to the stability threshold, then ifwe integrate (4.27) over the domain and use the no-flux boundary conditionwe conclude that\u2211Nj=1Cj = 0. However, we do not have this constraintwhen \u03bb 6= 0. This observation suggest that we need to split our analysis intoseveral cases.4.2.1 \u03bb 6= 0 and \u03bb \u223c O(1)First we introduce the reduced-wave Green function, which satisfies\u2206xG\u03c3(x, \u03be)\u2212 \u03c32G\u03c3(x, \u03be) = \u2212\u03b4(x\u2212 \u03be) , (4.28)\u2207xG\u03c3(x, \u03be) \u00b7 n = 0 , \u2200x \u2208 \u2202\u2126 , (4.29)G\u03c3(x, \u03be)\u2192 \u221212piln |x\u2212 \u03be|+R\u03c3(\u03be) , as x\u2192 \u03be . (4.30)We need an approximation to this Green\u2019s function when \u03c3 \u001c 1. Sincethere is no solution when \u03c3 = 0, then as we did in (2.3) we must seek G\u03c3444.2. Linear Stability Analysisfor \u03c3 \u001c 1 in the form of a singular asymptotic expansion asG\u03c3(x, \u03be) =1\u03c32|\u2126|+G0(x, \u03be) +O(\u03c32) , (4.31)where G0(x, \u03be) is the Neumann Green function defined in (4.9). Then if wedenote \u03c3 = \u03c4\u03bbD , we can represent the solution to (4.27) in the form\u03b7(x) = \u22122piN\u2211i=1CiG\u03c3(x,xi) = 2piN\u2211i=1Ci(\u2212D\u03c4\u03bb|\u2126|\u2212G0(x,xi) +O(\u03c32)).(4.32)By expanding this solution as x \u2192 xj and comparing the resulting ex-pression with the far-field behavior of the corresponding inner solution, weobtain the following matching condition near each spot:Cj\u03bd+Bj = \u22122piD\u03c4\u03bb|\u2126|N\u2211i=1Ci\u22122piR(xj)Cj\u2212\u2211i 6=j2piG0(xi,xj)Ci , 1 \u2264 j \u2264 N .(4.33)We then rewrite this system in matrix form by first introducing thenotationC \u2261\uf8eb\uf8ec\uf8ec\uf8edC1...CN\uf8f6\uf8f7\uf8f7\uf8f8 , B \u2261\uf8eb\uf8ec\uf8ec\uf8edB1...BN\uf8f6\uf8f7\uf8f7\uf8f8 , \u03a6 \u2261\uf8eb\uf8ec\uf8ec\uf8ed\u03a61...\u03a6N\uf8f6\uf8f7\uf8f7\uf8f8 , N \u2261\uf8eb\uf8ec\uf8ec\uf8edN1...NN\uf8f6\uf8f7\uf8f7\uf8f8 , (4.34)E \u22611NeeT =1N\uf8eb\uf8ec\uf8ec\uf8ed1 1 \u00b7 \u00b7 \u00b7 1.... . ....1 \u00b7 \u00b7 \u00b7 1\uf8f6\uf8f7\uf8f7\uf8f8 . (4.35)In terms of these new variables, (4.33) can be written in matrix form as(I +2pi\u03bdDN\u03c4\u03bb|\u2126|E + 2pi\u03bdG)C = \u2212\u03bdB . (4.36)From this system it follows that B is one order higher in \u03bd than Cwhen \u03bb \u223c O(1). Moreover, since the system (4.4) is linear, we may assume454.2. Linear Stability AnalysisN(y) \u223c O(1). This suggests that we introduce the asymptotic expansion as\u03a6j \u223c \u03bd(\u03a6j0 + \u03bd\u03a6j1 + \u00b7 \u00b7 \u00b7 ) , Nj \u223c Nj0 + \u03bdNj1 + \u00b7 \u00b7 \u00b7 , (4.37)Bj \u223c Bj0 + \u03bdBj1 + \u00b7 \u00b7 \u00b7 , Cj \u223c \u03bd(Cj0 + \u03bdCj1 + \u00b7 \u00b7 \u00b7 ) .Upon substituting this expansion into (4.25), and by using the results fromPrincipal Result 4, we obtain at leading order that\u2206\u03c1\u03a6j0 \u2212 (\u03bb+ 1)\u03a6j0 + 2fUj0Vj0\u03a6j0 + fU2j0Nj0 = 0 , \u2206\u03c1Nj0 = 0 , (4.38)\u03a6\u2032j0(0) = N\u2032j0(0) = 0 ; \u03a6j0 \u2192 0, Nj0 \u2192 Bj0 as \u03c1\u2192\u221e . (4.39)The solution of this system in terms of the linear operator L0 of Chapter 3,defined by L0\u03c6 = \u2206\u03c6\u2212 \u03c6+ 2\u03c9\u03c6, is simplyL0\u03a6j0 = \u03bb\u03a6j0 \u2212Bj0f\u03c720\u03c92 , Nj0 = Bj0 , (4.40)\u21d2 L0\u03a60 = \u03bb\u03a60 \u2212\u03c92f\u03c720B0 , N0 = B0 .Moreover, to leading order from the matching condition (4.36), we concludethat(I + \u00b5E)C0 = \u2212B0 , \u00b5 \u22612piD0N\u03c4\u03bb|\u2126|. (4.41)At the next order, we obtain from (4.25) that the equation for Nj1 is\u2206\u03c1Nj1 + (1\u2212 2Uj0Vj0)\u03a6j0 \u2212 U2j0Nj0 = 0 ,N \u2032j1(0) = 0 ; Nj1 \u2192 Cj0 ln \u03c1+Bj1 , as \u03c1\u2192\u221e .Upon integrating this equation, and using the divergence theorem, we obtainthe consistency condition thatCj0 =bf2\u03c720Bj0 \u2212\u222b \u221e0(1\u2212 2\u03c9f)\u03a6j0\u03c1 d\u03c1 , (4.42)\u21d2 C0 =bf2\u03c720B0 \u2212\u222b \u221e0(1\u2212 2\u03c9f)\u03a60\u03c1 d\u03c1 .464.2. Linear Stability AnalysisWe want to eliminate the\u222b\u221e0 \u03a60\u03c1 d\u03c1 term as before by integrating the equa-tion for \u03a60 in (4.40). This yields that(\u03bb+ 1)\u222b \u221e0\u03a60\u03c1d\u03c1 = 2\u222b \u221e0\u03c9\u03a60\u03c1 d\u03c1+bf\u03c720B0 . (4.43)Then, upon combining (4.41), (4.42) and (4.43), we conclude that[(1 + a)I + a\u00b5E ]B0 = \u22122(\u03bb+ 1\u2212 f)f(\u03bb+ 1)(I + \u00b5E)\u222b \u221e0\u03c9\u03a60\u03c1 d\u03c1 , (4.44)where a \u2261 b(\u03bb+1\u2212f)f2\u03c720(\u03bb+1). Using the fact that (I + kE)\u22121 = I \u2212 kk+1E , we getB0 = \u22122m(1 +bmf\u03c720)\u22121\uf8eb\uf8edI +\u00b5\u03c4\u03bb11 + bmf\u03c720(1 + \u00b5\u03c4\u03bb)E\uf8f6\uf8f8\u222b \u221e0\u03c9\u03a60\u03c1 d\u03c1 ,(4.45)where m is defined as m \u2261 1f \u22121\u03bb+1 . Upon substituting this expression backinto L0\u03a60 = \u03bb\u03a60 \u2212 \u03c92f\u03c720B0, we get a vector nonlocal eigenvalue problem(NLEP).Next, we will decompose \u03a60 into certain directions, most of which areunaffected by the matrix E . We observe that the geometric meaning of thematrix E is that of projecting a vector into the direction e = (1, 1, ..., 1)T .This suggests that we decompose \u03a60 into ke+ r, where k is some constantand r\u22a5e. Recall from (4.16) that the eigenvectors of the Neumann Greenmatrix G are such that 1\u221aNe, {qj}Nj=2 forms a orthonormal basis of RN andqj\u22a5e for j = 2, 3, ..., N . We decompose \u03a60 into this basis by writing\u03a60(|x|) = k(|x|)e+N\u2211j=2uj(|x|)qj , (4.46)where k(|x|), uj(|x|) are radially symmetric coefficient functions. Uponsubstituting this into the equation for \u03a60, and by using the formula for B0,474.2. Linear Stability Analysiswe obtain the two distinct NLEP\u2019sL0k(|x|)\u22122b(\u03bb+ 1\u2212 f)(\u03c4\u03bb+ \u00b5)(\u03bb+ 1)f2\u03c720\u03c4\u03bb+ b(1\u2212 f + \u03bb)(\u03c4\u03bb+ \u00b5)\u222b\u221e0 k\u03c9\u03c1 d\u03c1\u222b\u221e0 \u03c92\u03c1 d\u03c1\u03c92 = \u03bbk(|x|) ,L0uj \u22122b(\u03bb+ 1\u2212 f)(\u03bb+ 1)f2\u03c720 + b(1\u2212 f + \u03bb)\u222b\u221e0 uj\u03c9\u03c1 d\u03c1\u222b\u221e0 \u03c92\u03c1 d\u03c1\u03c92 = \u03bbuj . (4.47)These two distinct NLEP\u2019s are similar to those considered previously. Assuch we conclude that the component k(|x|) is linearly stable, but that thereis a stability threshold for uj(|x|) when\u03b2(\u03bb)|\u03bb=0 =2b(\u03bb+ 1\u2212 f)(\u03bb+ 1)f2\u03c720 + b(1\u2212 f + \u03bb)|\u03bb=0 = 1 ,which yields that\u03c720 =(1\u2212 f)bf2, Sc0 =\u221ab(1\u2212 f)f. (4.48)We remark here that at the stability threshold we have \u03bb = 0, whichseems to contradict our starting assumption \u03bb 6= 0. However, the previ-ous theorem of [16], states that when \u03b2(0) < 1 the NLEP has a positivereal eigenvalue. Another difficulty is that near the stability threshold theeigenvalue can be very small. In particular, if \u03bb = O(\u03bd), then the 2pi\u03bdDN\u03c4\u03bb|\u2126| Eterm in the matching condition (4.36) becomes the dominate term and theasymptotic expansions need a little modification. We will handle these twosituations in Section 4.2.2 and Section 4.2.3, respectively.4.2.2 \u03bb \u223c O(\u03bd) and \u03bb 6= 0Next we treat the case where \u03bb \u223c O(\u03bd) and \u03bb 6= 0, and we expand \u03bb as\u03bb = \u03bd\u03bb1 + \u03bd2\u03bb2 + .... For this case the results (4.27), (4.28), and (4.31)still hold. However, the only difference is that the D\u03c4\u03bb|\u2126| term is no longerO(\u03bd\u22121), but instead is O(\u03bd\u22122). This implies that the matching condition ismodified as (2piD0N\u03c4\u03bb1|\u2126|E + \u03bdI + 2pi\u03bd2G)C = \u2212\u03bd2B . (4.49)484.2. Linear Stability AnalysisSince the system is linear, then without loss of generality we may assume thatB \u223c O(\u03bd) and from (4.49) it seems that C \u223c O(\u03bd2). However, this scalingassumption would be inconsistent with the logarithmic growth condition inthe equation for Nj1. In fact, the leading-order solution to (4.49 is thatEC = 0, which is equivalent to C\u22a5e. We then expand the solutions as in(4.37) and obtain that the matching condition becomes(I + 2pi\u03bdG)C = \u2212\u03bdB . (4.50)Due to the properties of G in (4.16), it follows that B\u22a5e. Therefore, atleading order, we haveB0 = \u2212C0\u22a5e, L0\u03a60 = \u2212\u03c92f\u03c720B0, N0 = B0, \u21d2 \u03a60 = \u2212\u03c9f\u03c720B0 .(4.51)We may assume that C0 =\u2211Nj=2 djqj = \u2212B0, where qj are other eigenvec-tors of G that are orthogonal to e and dj are some constant coefficients. Atnext order, the equation for N1 is:\u2206N1 =\u03c9f\u03c720B0 \u2212\u03c92f2\u03c720B0 , (4.52)N \u20321(0) = 0 , N1 \u2192 C0 ln |y|+B1 , as |y| \u2192 \u221e .The solvability condition for this equation gives:C0 =bf\u03c720(1\u22121f)B0 = \u2212B0, \u21d2 \u03c70 =\u221ab(1\u2212 f)f2, S0 =\u221ab(1\u2212 f)f,(4.53)which is precisely the same threshold we obtained from (4.47). The solutionto (4.52) can then be written asN1 = B1 \u2212V1Pb(1\u2212 f)B0 = B1 +V1Pb(1\u2212 f)N\u2211j=2djqj . (4.54)494.2. Linear Stability AnalysisUpon substituting this expression into the equation for N2 we get\u2206N2 + (1\u22122\u03c9f)\u03a61 \u2212 2(\u03c9f\u03c70(\u03c71 +V1Pf2\u03c70) + \u03c70(\u2212\u03c71f\u03c720\u03c9 \u2212U1Pf3\u03c730))\u03a60\u2212\u03c92f2\u03c720N1 \u2212 2\u03c9f\u03c720(\u2212\u03c71f\u03c720\u03c9 \u2212U1Pf3\u03c730)N0 = 0 ,N \u20322(0) = 0 , N2 \u2192 C1 ln |y|+B2 , as |y| \u2192 \u221e .Upon using the divergence theorem on this equation for N2, and by using(4.51) and (4.54), we obtain that the following consistency condition musthold:C1 +\u222b \u221e0\u03a61\u03c1d\u03c1\u22122\u03c9f\u222b \u221e0\u03a61\u03c1 d\u03c1\u221211\u2212 fB1 \u22123b2(1\u2212 f)2\u222b \u221e0\u03c92V1P\u03c1 d\u03c1N\u2211j=2djqj \u2212 2f\u03c71b12 (1\u2212 f)32N\u2211j=2djqj = 0 . (4.55)Similarly, the matching condition (4.50) givesC1 + 2piGC0 = \u2212B1, \u21d2 C1 + 2piN\u2211j=2kjdjqj = \u2212B1. (4.56)The equation for \u03a61 then becomesL0\u03a61 +3f\u03c92V1pb2(1\u2212 f)2N\u2211j=2djqj +2f2\u03c71\u03c92(b(1\u2212 f))32N\u2211j=2djqj (4.57)+f\u03c92b(1\u2212 f)B1 = \u03bb1fb(1\u2212 f)\u03c9N\u2211j=2djqj .Upon integrating this equation and then substituting into (4.55) and (4.56),we get thatB1 = piN\u2211j=2djkjqj \u22121\u2212 ff\u222b \u221e0\u03c9\u03a61\u03c1 d\u03c1504.2. Linear Stability Analysis\u2212fb(1\u2212 f)N\u2211j=2djqj(b\u03c71(1\u2212 f)f\u03c70+32bf\u222b \u221e0\u03c92V1P\u03c1 d\u03c1+b2\u03bb1).Finally, we substitute this expression back into (4.57) to obtain the vectorNLEPL\u03a61 \u2261 L0\u03a61 \u2212\u222b\u221e0 \u03a61\u03c9\u03c1 d\u03c1\u222b\u221e0 \u03c92\u03c1 d\u03c1\u03c92 = \u03bb1(\u03c9 +f2\u2212 2f\u03c92)fb(1\u2212 f)N\u2211j=2djqj\u22123f\u03c92V1Pb2(1\u2212 f)2N\u2211j=2djqj \u2212fb(1\u2212 f)N\u2211j=2djqj(\u2212pikj +\u03c71\u03c70\u22123\u222b\u221e0 \u03c92V1P\u03c1 d\u03c12b2(1\u2212 f))\u03c92 .Upon decomposing \u03a61(x) = r(x)e+\u2211Nj=2 tj(x)qj as before, we obtain thatthe coefficient functions tj(x) satisfyLtj(x) =fdjb(1\u2212 f){\u03bb1(\u03c9 +f2\u2212 2f\u03c92)\u22123b(1\u2212 f)\u03c92V1P\u2212 \u03c92(\u2212pikj +\u03c71\u03c70\u221232b2(1\u2212 f)\u222b \u221e0\u03c92V1P\u03c1 d\u03c1)}.Finally, we use a solvability condition on this problem to calculate \u03bb1. FromSection 3.2 the adjoint operator L\u2217 has a one-dimensional nullspace \u03a8\u2217(x)in the class of radially symmetric functions, where \u03a8? = w + \u03c1w\u2032\/2 wasgiven in Section 3.2. We then impose the solvability condition as similar tothat done in Section 3.2 to conclude that\u222b \u221e0RHS \u00b7\u03a8\u2217\u03c1 d\u03c1 = 0 \u21d2 \u03bb1 = 2(1\u2212 f)(\u2212pikj +\u03c71\u03c70+3\u222b\u221e0 U1P\u03c1 d\u03c12(1\u2212 f)b2)= 2(1\u2212 f)(\u2212pikj \u2212S1S0+\u222b\u221e0 U1P\u03c1 d\u03c12(1\u2212 f)b2).With this expression we can calculate the next order term in the stabilitythreshold that makes \u03bb1 = 0 asSc1 = max2\u2264j\u2264N{S02(\u22122pikj +1(1\u2212 f)b2\u222b \u221e0U1P\u03c1 d\u03c1)} = max2\u2264j\u2264N{\u221ab(1\u2212 f)2f514.2. Linear Stability Analysis(\u22122pikj +1(1\u2212 f)b2((1\u2212 f)\u222b \u221e0U1QII\u03c1 d\u03c1\u2212\u222b \u221e0U1QI\u03c1 d\u03c1))} . (4.58)We remark here that since G is real symmetric, all of its eigenvalues kj arereal. Therefore, as expected, the stability threshold is real-valued.4.2.3 \u03bb=0Finally we consider the case where \u03bb = 0, which corresponding to the sta-bility threshold. The equation for \u03b7 now becomes\u2206\u03b7 = 2piN\u2211j=1Cj\u03b4(x\u2212 xj) . (4.59)As we mentioned before, if we integrate this PDE with the no-flux boundaryconditions on \u2202\u2126, we obtain thatN\u2211j=1Cj = 0 , \u03b7(x) = \u22122piN\u2211j=1CjG0(x,xj) , (4.60)where G0(x,xj) is the Neumann Green function defined as before. Usingthe same method in Section 3.3, we derive that \u2202\u2202SUj(y, Sj),\u2202\u2202SVj(y, Sj) aresolutions to (4.25) when \u03bb = 0. This leads to the relationBjCj=\u2202\u2202S\u03c7(Sj , f) . (4.61)Since we are assuming a common source strength Sj = S = Sc, (4.61) givesB = \u2202\u2202S\u03c7(Sc, f)C. We substitute this expression back into the matchingcondition (4.36) and, upon noticing that EC = 0, we have(I + \u03bd\u2202\u03c7\u2202S(Sc, f))C = \u22122pi\u03bdGC . (4.62)This implies that C is an eigenvector of G, and we obtain the relation1 + \u03bd\u2202\u03c7\u2202S(Sc, f) = \u22122pi\u03bdkj . (4.63)524.2. Linear Stability AnalysisAs we derived in (3.64) and (3.65), we calculate\u2202\u2202S\u03c7(Sc, f) = \u22121\u03bdb(1\u2212 f)f21S2c0+ (2b(1\u2212 f)f2Sc1S3c0\u2212\u222b\u221e0 U1P\u03c1 d\u03c1(1\u2212 f)b2) +O(\u03bd) .(4.64)We substitute this expression back into (3.63) and equate terms of a commonorder in \u03bd. In this way, we derive the stability threshold resultsSc0 =\u221ab(1\u2212 f)f, (4.65)Sc1 = max2\u2264j\u2264N{\u221ab(1\u2212 f)2f(\u22122pikj +1(1\u2212 f)b2((1\u2212 f)\u222b \u221e0U1QII\u03c1 d\u03c1\u2212\u222b \u221e0U1QI\u03c1 d\u03c1))} .Finally, by using (4.15) which relates S to D, we calculate the stabilitythreshold in terms of D. The results are summarized as follows:Principal Result 5. In the limit \u000f \u2192 0, and on the range D \u223c O( 1\u03bd ), themulti-spot patterns constructed in Principal Result 4 are linearly stable ifD < Dc =Dc0\u03bd+Dc1 + ... , (4.66)where Dc0 and Dc1 are defined byDc0 =f2E2|\u2126|24pi2N2(1\u2212 f)b, (4.67)Dc1 = min2\u2264j\u2264N{Dc0(2pikj \u22121(1\u2212 f)b2\u222b \u221e0U1P\u03c1 d\u03c1)} (4.68)= min2\u2264j\u2264N{Dc0(2pikj +1(1\u2212 f)b2\u222b \u221e0U1QI\u03c1 d\u03c1\u22121b2\u222b \u221e0U1QII\u03c1 d\u03c1)} .(4.69)We remark here that since we are solving for eigenvalues of a self-adjointoperator, all the eigenvalues are real-valued, and so the stability threshold isreal. Although we have discussed the three cases of \u03bb separately, the analysisindeed provides a uniform transition between the ranges of \u03bb. More specif-534.2. Linear Stability Analysisically, we obtain the same leading order results for the stability thresholdfrom (4.48), (4.53) and (4.65). Moreover, (4.58) also agrees with (4.65) atsecond order. Finally, in Section 4.2.2, we must have\u2211Nj=1Cj = 0 to havea solution, which also agrees with the solvability condition in Section 4.2.3.54Chapter 5Numerical ResultsIn this chapter, we perform some numerical experiments and compare theresults with the two term asymptotic approximations for the stability thresh-old derived in the previous chapter. For the periodic case, we will identifythe optimal lattice arrangement of spots. For the finite domain problem weillustrate our theory for the case of equally-spaced spots on a ring that isconcentric with the unit disk. For this finite domain problem, there is anexplicit formula for the Neumann Green function that will be used.5.1 Small S Asymptotics of \u03c7(S, f)For both the periodic and the finite domain problems, the same core prob-lems (3.3) and (4.4) arise in the asymptotic construction of the spot pattern.In these common inner problems, a key quantity is \u03c7(S, f), which appearsin the asymptotic boundary condition. Two-term asymptotic expansions for\u03c7(S, f) have been derived previously in Principal Result 1 and Principal Re-sult 4. We now solve the core problem numerically to compute the \u03c7(S, f),and we compare the numerical results for \u03c7(S, f) with the correspondingtwo-term asymptotic results derived in the small S limit.Since we are seeking radially symmetric solutions, solving the core prob-lem is actually solving an ODE system. We use the ODE boundary valueproblem solver BVP4C in Matlab. We now remark on a few details of thenumerical implementation.\u2022 The Laplace operator in R2 polar coordinates is expressed as:\u2206\u03c1 =d2d\u03c12+1\u03c1dd\u03c1.555.1. Small S Asymptotics of \u03c7(S, f)\u2022 Instead of solving the ODE systems on the whole interval [0,\u221e), weuse the interval [\u00b5,R], where R is a sufficiently large number so as toapproximate the infinite domain, while \u00b5 is a sufficiently small numberto avoid the singularity at r = 0. We chose R = 15 and \u00b5 = 0.005 inour computations.\u2022 Instead of using the boundary conditions for U(\u03c1) and V (\u03c1) as \u03c1\u2192\u221ein (3.4) directly, we set U(R) = 0 and V\u2032(R) \u223c SR . Then, after solvingthe core problem numerically, we define \u03c7(S, f) by \u03c7(S, f) = V (R)\u2212S lnR.\u2022 The boundary value solver requires a good initial guess consistent withthe boundary conditions. First we may want to use our small S asymp-totic approximation in Principal Result 1 as an initial guess. To dothis, we in principle need to know the radially symmetric solution of\u2206\u03c9\u2212\u03c9+\u03c92 = 0 in R2. However, this explicit solution is not availablein R2, and is only available in R1. As such, we adapt a homotopyalgorithm to find the initial guess. For a fixed f , we first start with asmall S0, and use the asymptotic approximation for the core problemin R. We then slowly increase the dimension from 1 to 2 and solve thecore problem using the previous step as an initial guess. After hav-ing obtained the core solution for S = S0 in R2 using this homotopystrategy, we then increase S and solve the core problem based on thepreviously computed solution.After computing the \u03c7(S, f) in this way, we compare the results withthe two term asymptotic expansions in (3.64). In the asymptotic approx-imation, we require numerical values for a few integrals. We obtain thatb \u2248\u222b\u221e0 \u03c9\u03c1d\u03c1 = 4.9343,\u222b\u221e0 U1QI\u03c1d\u03c1 \u2248 11.9131 and\u222b\u221e0 U1QI\u03c1 d\u03c1 \u2248 11.4384.Figure 5.1 shows results for f = 0.4 and f = 0.5, where the green (top)curves are the asymptotic approximations and the blue (bottom) curves arethe full numerical results. We observe that the two curves are rather closefor small S, which is what we should expect.565.2. Stability Threshold and the Optimal Lattice Arrangement0 1 2 3 4 5 6020406080100120140160180200\u03c7sf=0.40 1 2 3 4 50102030405060708090100\u03c7sf=0.5Figure 5.1: Numerical solution (bottom curves) and asymptotic results (topcurves) for \u03c7(S, f). In the left panel we fix f = 0.4, while f = 0.5 forthe right panel. In both pictures, the blue (bottom) curve is the numericalsolution while the green (top) one is the two term asymptotic expansion.5.2 Stability Threshold and the Optimal LatticeArrangementIn this section, we compute the stability threshold numerically and comparethe results with the two term asymptotic approximation.As derived in (3.63) and (3.65), the stability threshold for S, labeledby Sc, for the periodic spot problem is the largest S, corresponding to thesmallest D, that solves the transcendental equation\u2202\u2202S\u03c7(Sc, f) = \u22121\u03bd\u2212 2piR0,k +O(\u03bd) , (5.1)for some Bloch vector k in the first Brillouin zone. For the correspondingfinite domain problem, the stability threshold is the largest S that solves\u2202\u2202S\u03c7(Sc, f) = \u22121\u03bd\u2212 2piki , (5.2)for certain eigenvalues ki of the Neumann Green matrix.Since we have already computed \u03c7(S, f) above, we can use a cubic spline575.2. Stability Threshold and the Optimal Lattice Arrangementand a numerical derivative to get \u2202\u2202S\u03c7(S, f). Then we use a nonlinear equa-tion solver to compute the threshold directly.On the other hand, we have derived the two term asymptotic approxi-mation for the stability threshold in both cases. For the periodic case, wederived previously thatSc = \u03bd12\u221ab(1\u2212 f)f(1 + \u03bd(12b2\u222b \u221e0U1QII\u03c1 d\u03c1\u221212b2(1\u2212 f)\u222b \u221e0U1QI\u03c1 d\u03c1\u2212 pi mink\u2208\u2126\u2217\\0R0,k)) , (5.3)while for the finite domain problem we derived thatSc = \u03bd12\u221ab(1\u2212 f)f(1 + \u03bd(12b2\u222b \u221e0U1QII\u03c1 d\u03c1\u221212b2(1\u2212 f)\u222b \u221e0U1QI\u03c1 d\u03c1\u2212 pi min2\u2264i\u2264Nki)) . (5.4)Notice that the only key difference between these two expressions isthat mink\u2208\u2126\u2217\\0R0,k is replaced by min2\u2264i\u2264N ki. Therefore, we introduce aparameter c and our goal is to compare the solution to\u2202\u2202S\u03c7(Sc, f) = \u22121\u03bd\u2212 c , (5.5)with the expressionSc = \u03bd12\u221ab(1\u2212 f)f(1 + \u03bd(12b2\u222b \u221e0U1QII\u03c1 d\u03c1\u221212b2(1\u2212 f)\u222b \u221e0U1QI\u03c1 d\u03c1\u2212c2)),for some small \u000f. In Figure 5.2 we show numerical results that confirm thatthe full numerical results and asymptotic results agree rather well.Next, we identify the optimal lattice arrangement for the periodic case.As stated in Principal Result 3, the optimal lattice arrangement \u039bop withfixed primitive cell of area unity is the one which solve the following max-minproblem:arg max\u039b, |\u2126|=1{ mink\u2208\u2126\u2217\\0{R0,k}} (5.6)585.2. Stability Threshold and the Optimal Lattice Arrangement\u22121 \u22120.5 0 0.5 10.4270.4280.4290.430.4310.4320.433cS c\u22121 \u22120.5 0 0.5 10.6750.680.6850.690.6950.70.7050.710.7150.72cS cFigure 5.2: Numerical solution to (5.5) and the two-term asymptotic ap-proximations for Sc with different c. Left panel: f = 0.4 and \u000f = 0.01.Right panel: f = 0.5 and \u000f = 0.05. The blue (top) curve is the numericalsolution while the green (bottom) one is the asymptotic approximations inboth cases.Therefore, if we want to find the optimal lattice numerically, we need toknow how to calculate R0,k. First we follow the process in [2] and [8], toderive an explicit expression for the Bloch Green function and its regularpart R0,k. Recall that the Bloch Green function satisfies:\u2206G0,k(x) = \u2212\u03b4(x), \u2200x \u2208 \u2126 ,G0,k(x+Li) = e\u2212ik\u00b7LiG0,k(x) , \u2200x \u2208 d\u2212i ,\u2202n\u2212G0,k(x+Li) = e\u2212ik\u00b7Li\u2202n+G0,k(x) , \u2200x \u2208 d\u2212i .The free space Green\u2019s function in the absence of any boundary conditionsis Gfree(x) = \u2212 12pi ln |x|. We then observe that the infinite sumG(x) =\u2211l\u2208\u039bGfree(x+ l)eik\u00b7l ,satisfies the PDE together with the quasi-periodic boundary conditions. To595.2. Stability Threshold and the Optimal Lattice Arrangementverify that it satisfies these boundary conditions we calculateG(x+Li) =\u2211l\u2208\u039bGfree(x+Li + l)eik\u00b7l ,=\u2211Li+l\u2208\u039bGfree(x+ (Li + l))eik\u00b7(Li+l)eik\u00b7(\u2212Li) ,= e\u2212ik\u00b7Li\u2211l\u2032\u2208\u039bGfree(x+ l\u2032)eik\u00b7l\u2032= e\u2212ik\u00b7LiG(x) .The second line above follows since Li \u2208 \u039b and l\u2032= Li + l.By the Poisson summation formula proved in (2.4) in Chapter 2, andthe fact that G\u0302free(\u03be) = 1|\u03be|2 and |\u2126| = 1, we haveG(x) =\u2211l\u2208\u039bGfree(x+ l)eik\u00b7l =\u2211d\u2208\u039b\u2217G\u0302free(d\u2212 k)eix\u00b7(d\u2212k) =\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2.It is easy to prove from an integral test that the last series is not absolutelyconvergent. However, we can show it is actually conditionally convergent forx 6= 0 by decomposing it into two parts as was done in [2]. We pick some \u03b7in \u03b7 \u2208 (0, 1), and rewrite the infinite series as\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2=\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2(1\u2212 e\u2212 |d\u2212k|24\u03b72 + e\u2212 |d\u2212k|24\u03b72 ) ,=\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2e\u2212 |d\u2212k|24\u03b72 +\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2(1\u2212 e\u2212 |d\u2212k|24\u03b72 ) .The first term is an absolutely convergent series, which we denote asGFourier(x) =\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2e\u2212 |d\u2212k|24\u03b72 . (5.7)We claim that the second term equals another absolutely convergent seriesover the original lattice \u039b. We can write this series in a convenient formusing the following lemma:605.2. Stability Threshold and the Optimal Lattice ArrangementLemma 5.\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2(1\u2212 e\u2212 |d\u2212k|24\u03b72 ) =\u2211l\u2208\u039bFsing(x+ l)eik\u00b7l , \u2200x \u2208 \u2126\\0 , (5.8)where Fsing(x) \u2261 12pi\u222b\u221eln(2\u03b7) e\u2212 |x|24 e2sds = 14piE1(\u03b72|x|2), and E1(z) is theexponential integral defined by E1(z) =\u222b\u221ez t\u22121e\u2212t dt.Proof. Firstly, we observe that 12pi\u222b\u221eln(2\u03b7) e\u2212 |x|24 e2sds = 14piE1(\u03b72|x|2) by us-ing a simple change of variables. Then, according to [1], the exponentialintegral E1(z) has the decay property, E1(z) < e\u2212z ln(1 + 1z ), so that the se-ries over \u039b given by the right hand-side of (5.8) converges absolutely. Then,by using the Poisson summation formula as proved in (2.4), we get\u2211l\u2208\u039bFsing(x+ l)eik\u00b7l =\u2211d\u2208\u039b\u2217F\u0302sing(d\u2212 k)eix\u00b7(d\u2212k) . (5.9)Upon comparing this result with the statement that we want to prove, weneed only show that F\u0302sing(\u03be) = 1|\u03be|2 (1 \u2212 e\u2212 |\u03be|24\u03b72 ). To prove this we showthat the inverse Fourier transform of the right hand-side is Fsing(x). No-tice that both Fsing(x) and the right hand-side are radially symmetric,and that the inverse Fourier transform of a radially symmetric functionis the inverse Hankel transform of order zero (cf. [11]), so that f(r) =(2pi)\u22121\u222b\u221e0 f\u02c6(\u03c1)J0(\u03c1r)\u03c1 d\u03c1. Upon using the well-known inverse Hankel trans-form (cf. [11]) \u222b \u221e0e\u2212\u03c12e\u22122s\u03c1J0(\u03c1r) d\u03c1 =12e2s\u2212r2 e2s4 ,and the fact that1\u03c12(1\u2212 e\u2212 \u03c124\u03b72 ) = 2\u222b \u221eln(2\u03b7)e\u2212\u03c12e\u22122s\u22122sds ,we calculate the inverse Fourier transform of 1\u03c12 (1\u2212 e\u2212 \u03c124\u03b72 ) as12pi\u222b \u221e0(1\u03c12(1\u2212 e\u2212 \u03c124\u03b72 ))J0(\u03c1r)\u03c1 d\u03c1 =12pi\u222b \u221e0(\u222b \u221eln(2\u03b7)2e\u2212\u03c12e\u22122s\u22122s ds)J0(\u03c1r)\u03c1 d\u03c1615.2. Stability Threshold and the Optimal Lattice Arrangement=1pi\u222b \u221eln(2\u03b7)e\u22122s(\u222b \u221e0e\u2212\u03c12e\u22122s\u03c1J0(\u03c1r) d\u03c1)ds =12pi\u222b \u221eln(2\u03b7)e\u22122se2s\u2212r24 e2sds=12pi\u222b \u221eln(2\u03b7))e\u2212r24 e2sds .Thus, we conclude that F\u22121( 1|\u03be|2 (1 \u2212 e\u2212 |\u03be|24\u03b72 )) = Fsing(x), which completesthe proof of the lemma.We remark here that the series over the reciprocal lattice \u039b\u2217, given bythe left hand-side of (5.8), is only conditionally convergent, while the seriesover the original lattice \u039b, given by the right hand-side of (5.8), convergesabsolutely and we denote it byGspatial(x) =\u2211l\u2208\u039bFsing(x+ l)eik\u00b7l . (5.10)In this way, we have an explicit expression for the Bloch Green func-tion G0,k = G(x), and have separated it into the sum of two absolutelyconvergent series asG(x) = GFourier(x) +Gspatial(x) . (5.11)We remark here that strictly speaking the demonstration above is not com-pletely rigorous. The Poisson summation formula proved previously requiresthat the function to be in L1, but Gfree(x) is not. The way to circumventthis technical difficulty is to first define the two absolutely convergent se-ries GFourier(x) and Gspatial(x) as in (5.7) and (5.10). Then, we defineG(x) = GFourier(x) + Gspatial(x) and simply prove it satisfies the differen-tial equation and the quasi periodic boundary conditions. Notice that G(x)is independent of the choice of \u03b7 since \u2200x \u2208 \u2126\\0,GFourier(x) +Gspatial(x) =\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2e\u2212 |d\u2212k|24\u03b72 +eix\u00b7(d\u2212k)|d\u2212 k|2(1\u2212 e\u2212 |d\u2212k|24\u03b72 ) ,=\u2211d\u2208\u039b\u2217eix\u00b7(d\u2212k)|d\u2212 k|2,625.2. Stability Threshold and the Optimal Lattice Arrangementwhich is independent of \u03b7.Next we calculate the singular behaviour of G(x) as x \u2192 0. The termGFourier(0) is finite so this term is readily calculated. However, in the seriesGspatial(x), there is a singularity as x \u2192 0 for the term corresponding tol = 0, owing to the fact that the exponential integral E1(z) has a singularityat 0. Upon using the well-known series expansion of E1(z)E1(z) = \u2212\u03b3 \u2212 ln(z)\u2212\u221e\u2211n=1(\u22121)nznnn!, | arg z| < pi , (5.12)as given in \u00a75.1.11 of [1], where \u03b3 = 0.57721 \u00b7 \u00b7 \u00b7 is Euler\u2019s constant, we derivethatFsing(x) \u223c \u2212\u03b34pi\u2212ln \u03b72pi\u2212ln |x|2pi+ o(1), as x\u2192 0. (5.13)This shows that G(x) has the expected logarithmic singularity as x \u2192 0,and that the regular part of the Bloch Green\u2019s function isR0,k = limx\u21920(G(x) +12piln |x|), (5.14)=\u2211d\u2208\u039b\u22171|d\u2212 k|2e\u2212 |d\u2212k|24\u03b72 +\u2211l\u2208\u039b\\0eik\u00b7lFsing(l)\u2212\u03b34pi\u2212ln \u03b72pi.We remark here that if we take conjugate of this expression, we get thesame quantity due to the symmetry of the lattice. This gives an alternativeproof that R0,k is real-valued. In addition, since R0,k only depends on \u2126and k, the expression above should be independent of the choice of \u03b7. Toestablish this result, we take the derivative of (5.14) with respect to \u03b7 andprove it vanishes. Upon differentiating (5.14), we obtain\u2202\u2202\u03b7R0,k =\u2211d\u2208\u039b\u221712\u03b73e\u2212 |d\u2212k|24\u03b72 \u221212pi\u03b7\u2211l\u2208\u039b\\0e\u2212|l|2\u03b72eik\u00b7l \u221212pi\u03b7. (5.15)635.2. Stability Threshold and the Optimal Lattice ArrangementTo show that this expression vanishes, it is equivalent to show that\u2211d\u2208\u039b\u2217pi\u03b72e\u2212 |x\u2212d|24\u03b72 = 1 +\u2211l\u2208\u039b\\0e\u2212|l|2\u03b72eix\u00b7l =\u2211l\u2208\u039be\u2212|l|2\u03b72eix\u00b7l. (5.16)Notice that the left hand-side is an absolutely convergent series and anintegrable function due to the exponential decay. Thus, as we have shown in(2.2), it can be decomposed into a Fourier series of eix\u00b7l, where l \u2208 (\u039b\u2217)\u2217 = \u039band the coefficient of eix\u00b7l is calculated as1|\u2126\u2217|\u222b\u2126\u2217f(y)e\u2212iy\u00b7l dy =1|\u2126\u2217|\u222b\u2126\u2217(\u2211d\u2208\u039b\u2217pi\u03b72e\u2212 |y\u2212d|24\u03b72 )e\u2212iy\u00b7l dy(\u2200d \u2208 \u2126\u2217, \u2200l \u2208 \u2126, l \u00b7 d = 2kpi, k \u2208 Z) =1|\u2126\u2217|\u222bR2pi\u03b72e\u2212 |y|24\u03b72 e\u2212iy\u00b7l dy(the Fourier transform of a Gaussian) =1|\u2126\u2217|pi\u03b724pi\u03b72e\u2212|l|2\u03b72(|\u2126| = 1, then |\u2126\u2217| = 4pi2) = e\u2212|l|2\u03b72 .This establishes that \u2202\u2202\u03b7R0,k = 0, which yields that R0,k is independent of\u03b7.The explicit expression (5.14) provides a way to calculate R0,k numer-ically. Since the two series converge absolutely, for a fixed lattice \u039b with|\u2126| = 1, we can truncate \u039b, \u039b\u2217 by a finite subset to get a good approxi-mation of R0,k. We then minimize it numerically over k \u2208 \u2126\u2217\\0. Noticethat Lemma 2 is useful here since it tells us that R0,k blows up as k \u2192 0,thus we can minimize it away from 0. Then, we change the lattice andmaximize R(\u039b) \u2261 mink\u2208\u2126\u2217\\0{R0,k} over different lattices with |\u2126| = 1. Thenumerical results shown in [8] indicates that R(\u039b) is maximized for a regularhexagonal lattice \u039bop and that R(\u039b\u2217) = \u22120.079124. For a regular hexagonallattice, Table 5.1 compares the numerical results for the stability threshold,measured in terms of the source strength, and the corresponding one- andtwo-term asymptotic approximations.645.3. Case Study: N Peaks on a RingLattice Sc Leading order Two term approximation\u000f = 0.1 1.3854 1.3603 1.3706\u000f = 0.01 0.4306 0.4302 0.430526Table 5.1: Source strength threshold and its asymptotic approximation fora regular hexagonal lattice with |\u2126| = 1 and f = 0.4.5.3 Case Study: N Peaks on a RingIn this section, we implement our stability theory for the finite domain fora particular arrangement of spots inside the unit disk \u2126 = D1. We take 5points {xi}5i=1 equally distributed on a circle of radius 0.5 concentric withinthe unit disk, as shown in Figure 5.3. The centers of the localized spotscorresponds to the locations of these points.Figure 5.3: 5 localized spots on a ring concentric within the unit disk.For this special symmetric configuration of 5 equally-spaced spots ona ring, the corresponding Neumann Green matrix G has a constant rowsum, which implies that e is an eigenvector of G. This spot configurationconsisting of equally-spaced spots is one of the simplest ways to ensure that655.3. Case Study: N Peaks on a Ringe is an eigenvector of G.For the unit disk, there is an explicit formula for the Neumann Green\u2019sfunction G0(x, \u03be) and its regular part. As derived in [9], we haveG0(x, \u03be) =12pi(\u2212 ln |x\u2212 \u03be| \u2212 ln ||\u03be|x\u22121|\u03be|\u03be|+12(|x|2 + |\u03be|2)\u221234) , (5.17)and thus the regular part R0(x) is given byR0(x) =12pi(\u2212 ln ||x|x\u22121|x|x|+ |x|2 \u221234) . (5.18)Without loss of generality we label the spot locations xi on the ringas xi = (12 cos2pi(i\u22121)5 ,12 sin2pi(i\u22121)5 )T , for i = 1, 2, ..., 5. We then substitutethis into (5.17) and (5.18) to obtain G. By using Matlab, we numericallycalculate all of the eigenvalues of G as k1 = \u22120.2126, k2 = k3 = 0.1392,and k4 = k5 = \u22120.1174. Next, we choose the smallest eigenvalue other thanthe one that corresponding to e. This is the eigenvalue k4 = k5 = \u22120.1174,which we then use to calculate the stability threshold in terms of the sourcestrength. This allows us to numerically evaluate the second order term inthe stability threshold. The full numerical results for the stability threshold,measured in terms of the source strength, are compared versus the one- andtwo-term asymptotic results in Table 5.2.Lattice Sc Leading order Two term approximation\u000f = 0.05 0.9742 0.9713 0.9619\u000f = 0.02 0.6110 0.6083 0.6107Table 5.2: The stability threshold in terms of the source strength S and itsone- and two-term asymptotic approximation for a 5 spot pattern on a ringof radius 0.5 concentric within the unit disk with f = 0.4.We remark that in this case, there is an analytical way to determine theeigenvalues of the Neumann Green matrix G. Since this matrix is cyclic,its eigenvectors are qi = (1, \u03c9i\u22121, (\u03c9i\u22121)2, ..., (\u03c9i\u22121)n\u22121)T , for i = 1, 2, ..., n,while the corresponding eigenvalue is f(\u03c9i\u22121), where \u03c9 is the n-th root of665.3. Case Study: N Peaks on a Ringunity. Here f(x) =\u2211nk=1 ckxk\u22121 and (c1, c2, ..., cn) is the first row of G.We can calculate the eigenvalues in this way and obtain the same results asgiven above by a direct numerical calculation of the eigenvalues by Matlab.67Chapter 6SummaryIn this thesis we have studied the linear stability of steady-state localizedspot patterns for a singularly perturbed Brusselator reaction-diffusion sys-tem in both a periodic and finite domain setting. For both problems, thereis a stability threshold Dc \u223c O(\u2212 1ln \u000f) that characterizes a zero eigenvaluecrossing. We have calculated a two term asymptotic approximation forDc through an asymptotic solution of a singularly perturbed linear eigen-value problem. In the periodic setting, we first use Floquet-Bloch theory toconvert the whole plane problem into a problem posed on a primitive celltogether with the Bloch boundary conditions. Then we obtain the leadingorder approximation for Dc by analyzing a leading order nonlocal eigenvalueproblem (NLEP) derived using the method of matched asymptotic expan-sions. This leading order NLEP is independent of the geometry of the lattice\u039b and the Bloch vector k. In order to characterize the effect of the latticeand the Bloch vector on the stability threshold, we calculated a higher or-der approximation for Dc by imposing a solvability condition to the nextorder equations. The calculation leads to a formula for a real-valued contin-uous band of spectra of the linearization that lies within a small ball nearthe origin in the spectral plane when D is near the leading order stabilitythreshold. The refined approximation to the stability threshold is obtainedfrom the requirement that this band of spectrum lies in the left half of thespectral plane. The correction to the leading order stability threshold ob-tained in this way depends on the regular part R0,k of the Bloch Greenfunction, which in turn is determined by the lattice and the Bloch vector k.An explicit formula for R0,k is also derived for numerical computation usingEwald summation methods. This formula is used to determine the optimallattice arrangement which allows for the largest stability threshold.68Chapter 6. SummaryThe analysis for the finite domain problem is similar, with the key dif-ference being that the N spots interact with each other through a NeumannGreen matrix G. For a pattern with arbitrarily-located spots, this leads usto analyze N distinct problems, one near each of the spots. For simplicity,we restrict the locations of the spots so that the spots have a common sourcestrength. In this way, the local problem near each of the spots is the same.By decomposing the solution to the linearized problem into the directionsof the eigenvectors of G, the analysis becomes very similar to that for theperiodic problem. More specifically, we obtain the leading order approxima-tion for the stability threshold through an NLEP that is the same in eachof N \u2212 1 directions. We then calculate the second order approximation byimposing a solvability condition in each direction on the second order terms.This higher order approximation to the stability threshold depends on thematrix eigenvalues of G.For both the periodic and finite domain problems, we also provide a quickway to derive the stability threshold, which avoids any detailed calculationof spectra near the origin in the spectral plane. This simplified analysisshows that the stability threshold can be determined by solving a nonlinearequation. Numerical comparison between the two-term approximation forthe stability threshold in terms of the source strength Sc and the resultsobtained from solving the nonlinear equation is provided. For the finitedomain problem we illustrate our theory for a case study of N = 5 equally-spaced localized spots on a circular ring that is concentric with the unitdisk.There are some open problems suggested by this study. Firstly, forthe periodic case, numerical evidence obtained from computing the regularpart of the Bloch Green\u2019s function indicates that it is the regular hexago-nal lattice that offers the optimum stability threshold. However, it wouldbe preferable to obtain a rigorous analytic proof of this result. Secondly,although we have employed a systematic asymptotic method to calculate arefined approximation to the stability threshold for the finite domain prob-lem, it would be interesting to extend the rigorous leading-order analysis in[16] to rigorously derive the second order term for the stability threshold.69Chapter 6. SummaryThirdly, it would be interesting to try to extend the rigorous frameworkof [16] to rigorously analyze the periodic problem. The technical difficultyhere is that, in contrast to the finite domain problems considered in [16] thathave discrete spectra, the periodic problem requires analyzing the edges ofa band of continuous spectra. Fourthly, it would be interesting to give aprecise relationship between the stability threshold for a multi-spot patternwith regularly spaced spots on a very large but finite domain and that forthe periodic problem. It is expected that the stability thresholds for thesetwo problems would be similar, with the only difference being essentiallythe perturbing effect of a distant domain boundary. More specifically, uponcomparing the stability thresholds for the periodic and finite domain prob-lems, we identify a formal correspondence that the regular part of the BlochGreen function R0,k is replaced by the eigenvalues ki of the Neumann Greenmatrix. Since we may view the periodic case as the limit of a truncatedlattice, i.e. \u039bN = {n1l1 + n2l2||ni| \u2264 N, i = 1, 2}, the question then ishow do the matrix eigenvalues ki approximate, or discretize, the continuousband R0,k? Finally, we remark that the analysis in this thesis has focusedon determining refined formulae for the stability thresholds associated withO(1) eigenvalues that result from zero eigenvalue crossings. However, it iswell-known that there are additional small eigenvalues of order \u03bb \u223c O(\u000f2)that are associated with the translation modes. Unstable eigenvalues in thisclass lead to weak instabilities that are only manifested over very long timeintervals. It would be interesting to calculate the stability thresholds forthese eigenvalues for both the periodic and finite domain problems. For thefinite domain problem, a leading order analysis of these eigenvalues is givenin [16] for a related Gierer-Meinhardt reaction-diffusion system.70Bibliography[1] M. Abramowitz, I. A. Stegun, et al. Handbook of mathematical func-tions, volume 1. Dover New York, 1972.[2] G. Beylkin, C. Kurcz, and L. Monzo\u00b4n. Fast algorithms for helmholtzgreen\u2019s functions. Proceedings of the Royal Society A: Mathematical,Physical and Engineering Science, 464(2100):3301\u20133326, 2008.[3] W. Chen and M. J. Ward. The stability and dynamics of localizedspot patterns in the two-dimensional gray-scott model. arXiv preprintarXiv:1009.2805, 2010.[4] X. Chen and Y. Oshita. An application of the modular function in non-local variational problems. Archive for Rational Mechanics and Analy-sis, 186(1):109\u2013132, 2007.[5] B. Gidas, W. M. Ni, and L. Nirenberg. Symmetry of positive solutionsof nonlinear elliptic equations in Rn. Adv. Math. Suppl. Stud. A, 7:369\u2013402, 1981.[6] A. Gierer and H. Meinhardt. A theory of biological pattern formation.Kybernetik, 12(1):30\u201339, 1972.[7] P. Gray and S. K. Scott. Sustained oscillations and other exotic pat-terns of behaviour in isothermal reactions. The Journal of PhysicalChemistry, 89(1):22\u201332, 1985.[8] D. Iron, John R., M. J. Ward, and J. Wei. Logarithmic expansions andthe stability of periodic patterns of localized spots for reaction-diffusionsystems in R2. Journal of nonlinear science, submitted.71Bibliography[9] T. Kolokolnikov, M. S. Titcombe, and M. J. Ward. Optimizing thefundamental neumann eigenvalue for the laplacian in a domain withsmall traps. European Journal of Applied Mathematics, 16(02):161\u2013200, 2005.[10] T. Kolokolnikov, M. J. Ward, and J. Wei. Spot self-replication anddynamics for the schnakenburg model in a two-dimensional domain.Journal of nonlinear science, 19(1):1\u201356, 2009.[11] R. Piessens. The hankel transform. The Transforms and ApplicationsHandbook, 2:9\u20131, 2000.[12] I. Prigogine and R. Lefever. Symmetry breaking instabilities in dissi-pative systems. ii. The Journal of Chemical Physics, 48(4):1695\u20131700,1968.[13] I. Rozada, S. J. Ruuth, and M. J. Ward. The stability of localized spotpatterns for the brusselator on the sphere. SIAM Journal on AppliedDynamical Systems, 13(1):564\u2013627, 2014.[14] A. M. Turing. The chemical basis of morphogenesis. PhilosophicalTransactions of the Royal Society of London. Series B, Biological Sci-ences, 237(641):37\u201372, 1952.[15] V. K. Vanag and I. R. Epstein. Localized patterns in reaction-diffusionsystems. Chaos: An Interdisciplinary Journal of Nonlinear Science,17(3):037110, 2007.[16] J. Wei and M. Winter. Spikes for the two-dimensional gierer-meinhardtsystem: the weak coupling case. Journal of Nonlinear Science,11(6):415\u2013458, 2001.[17] M. Winter and J. Wei. Existence, classification and stability analysisof multiple-peaked solutions for the gierer-meinhardt system in R1.Methods and Applications of Analysis, 14(2):119\u2013164, 2007.72","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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