Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems in $\R^2
aa r X i v : . [ n li n . PS ] D ec Under consideration for publication in the Journal of Nonlinear Science Logarithmic Expansions and the Stability of PeriodicPatterns of Localized Spots for Reaction-Diffusion Systemsin R D. IRO N, J. RUMSE Y, M. J. WARD, and J. WE I
David Iron; Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada,John Rumsey; Faculty of Management, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada,Michael Ward; Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada,Juncheng Wei, Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada andDepartment of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong. ( Received 5 June 2018 ) The linear stability of steady-state periodic patterns of localized spots in R for the two-component Gierer-Meinhardt(GM) and Schnakenburg reaction-diffusion models is analyzed in the semi-strong interaction limit corresponding to anasymptotically small diffusion coefficient ε of the activator concentration. In the limit ε →
0, localized spots in theactivator are centered at the lattice points of a Bravais lattice with constant area | Ω | . To leading order in ν = − / log ε ,the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies D = D /ν , for some D independent of the lattice and the Bloch wavevector kkk . From a combination of the method ofmatched asymptotic expansions, Floquet-Bloch theory, and the rigorous study of certain nonlocal eigenvalue problems,an explicit analytical formula for the continuous band of spectrum that lies within an O ( ν ) neighborhood of the origin inthe spectral plane is derived when D = D /ν + D , where D = O (1) is a de-tuning parameter. The periodic pattern islinearly stable when D is chosen small enough so that this continuous band is in the stable left-half plane Re( λ ) < kkk . Moreover, for both the Schnakenburg and GM models, our analysis identifies a model-dependent objective function,involving the regular part of the Bloch Green’s function, that must be maximized in order to determine the specificperiodic arrangement of localized spots that constitutes a linearly stable steady-state pattern for the largest value of D .From a numerical computation, based on an Ewald-type algorithm, of the regular part of the Bloch Green’s functionthat defines the objective function, it is shown within the class of oblique Bravais lattices that a regular hexagonal latticearrangement of spots is optimal for maximizing the stability threshold in D . Key words: singular perturbations, localized spots, logarithmic expansions, Bravais lattice, Floquet-Bloch theory,Green’s function, nonlocal eigenvalue problem.
Spatially localized spot patterns occur for various classes of reaction-diffusion (RD) systems with diverse applicationsto theoretical chemistry, biological morphogenesis, and applied physics. A survey of experimental and theoreticalstudies, through RD modeling, of localized spot patterns in various physical or chemical contexts is given in [ ].Owing to the widespread occurrence of localized patterns in various scientific applications, there has been considerablefocus over the past decade on developing a theoretical understanding of the dynamics and stability of localizedsolutions to singularly perturbed RD systems. A brief survey of some open directions for the theoretical study oflocalized patterns in various applications is given in [ ]. More generally, a wide range of topics in the analysis offar-from-equilibrium patterns modeled by PDE systems are discussed in [ ].In this broad context, the goal of this paper is to analyze the linear stability of steady-state periodic patternsof localized spots in R for two-component RD systems in the semi-strong interaction regime characterized by an D. Iron, J. Rumsey, M. J. Ward, J. Wei asymptotically large diffusivity ratio. For concreteness, we will focus our analysis on two specific models. One modelis a simplified Schnakenburg-type system(1.1) v t = ε ∆ v − v + uv , τ u t = D ∆ u + a − ε − uv , where 0 < ε ≪ D > τ >
0, and a >
0, are parameters. The second model is the prototypical Gierer-Meinhardt(GM) model formulated as(1.2) v t = ε ∆ v − v + v /u , τ u t = D ∆ u − u + ε − v , where 0 < ε ≪ D >
0, and τ >
0, are parameters.Our linear stability analysis for these two models will focus on the semi-strong interaction regime ε → D = O (1). For ε →
0, the localized spots for v are taken to be centered at the lattice points of a general Bravaislattice Λ, where the area | Ω | of the primitive cell is held constant. A brief outline of lattices and reciprocal lattices isgiven in § D . Through a numerical computation of this objective function we will show that itis a regular hexagonal lattice arrangement of spots that yields this optimal stability threshold.For the corresponding problem in 1-D, the stability of periodic patterns of spikes for the GM model was analyzed in[ ] by using the geometric theory of singular perturbations combined with Evans-function techniques. On a bounded1-D domain with homogeneous Neumann boundary conditions, the stability of N -spike steady-state solutions wasanalyzed in [ ] and [ ] through a detailed study of certain nonlocal eigenvalue problems. On a bounded 2 − D domainwith Neumann boundary conditions, a leading order in ν = − / log ε rigorous theory was developed to analyze thestability of multi-spot steady-state patterns for the GM model (cf. [ ], [ ]), the Schnakenburg model (cf. [ ]),and the Gray-Scott (GS) model (cf. [ ]), in the parameter regime where D = D /ν ≫
1. For the Schnakenburg andGM models, the leading-order stability threshold for D corresponding to a zero eigenvalue crossing was determinedexplicitly. A hybrid asymptotic-numerical theory to study the stability, dynamics, and self-replication patterns ofspots, that is accurate to all powers in ν , was developed for the Schnakenburg model in [ ] and for the GS model in[ ]. In [ ] and [ ], the stability and self-replication behavior of a one-spot solution for the GS model was analyzed.One of the key features of the finite domain problem in comparison with the periodic problem is that the spectrumof the linearization of the former is discrete rather than continuous. As far as we are aware, to date there hasbeen no analytical study of the stability of periodic patterns of localized spots in R on Bravais lattices for singularlyperturbed two-component RD systems. In the weakly nonlinear Turing regime, an analysis of the stability of patternson Bravais lattices in R using group-theoretic tools of bifurcation theory with symmetry was done in [ ] and [ ].By using the method of matched asymptotic expansions, in the limit ε → R . The stability of this solution with respect to O (1) time-scale instabilities arisingfrom zero eigenvalue crossings is then investigated by first using the Floquet-Bloch theorem (cf. [ ], [ ]) to formulatea singularly perturbed eigenvalue problem in the Wigner-Seitz cell Ω with quasi-periodic boundary conditions on ∂ Ωinvolving the Bloch vector kkk . In § § §
4, the spectrum of the linearized eigenvalue problemis analyzed by using the method of matched asymptotic expansions combined with a spectral analysis based onperturbations of a nonlocal eigenvalue problem. More specifically, to leading-order in ν = − / log ε it is shown that a he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems D ∼ D /ν , where D is a constant that depends on the parameters in the RDsystem, but is independent of the lattice geometry except through the area | Ω | of the Wigner-Seitz cell. Therefore,to leading-order in ν , the stability threshold is the same for any periodic spot pattern on a Bravais lattice Λ when | Ω | is held fixed. In order to determine the effect of the lattice geometry on the stability threshold, an expansionto higher-order in ν must be undertaken. In related singularly perturbed eigenvalue problems for the Laplacian in2-D domains with holes, the leading-order eigenvalue asymptotics in the limit of small hole radius only dependson the number of holes and the area of the domain, and not on the arrangement of the holes within the domain.An analytical theory to calculate higher order terms in the eigenvalue asymptotics for these problems, which haveapplications to narrow-escape and capture phenomena in mathematical biology, is given in [ ], [ ], and [ ].To determine a higher-order approximation for the stability threshold for the periodic spot problem we performa more refined perturbation analysis in order to calculate the continuous band λ ∼ νλ ( kkk, D , Λ) of spectra thatlies within an O ( ν ) neighborhood of the origin, i.e that satisfies | λ ( kkk, D , Λ) | ≤ O ( ν ), when D = D /ν + D forsome de-tuning parameter D = O (1). This band is found to depend on the lattice geometry Λ through the regularpart of certain Green’s functions. For the Schnakenburg model, λ depends on the regular part R b ( kkk )of the BlochGreen’s function for the Laplacian, which depends on both kkk and the lattice. For the GM Model, λ depends on both R b ( kkk ) and the regular part R p of the periodic source-neutral Green’s function on Ω. For both models, this band ofcontinuous spectrum that lies near the origin when D − D /ν = O (1) is proved to be real-valued.For both the Schnakenburg and GM models, the de-tuning parameter D on a given lattice is chosen so that λ < kkk . Then, to determine the lattice for which the steady-state spot pattern is linearly stable for thelargest possible value of D , we simply maximize D with respect to the lattice geometry. In this way, for each ofthe two RD models, we derive a model-dependent objective function in terms of the regular parts of certain Green’sfunctions that must be maximized in order to determine the specific periodic arrangement of localized spots that islinearly stable for the largest value of D . The calculation of the continuous band of spectra near the origin, and thederivation of the objective function to be maximized so as to identify the optimal lattice, is done for the Schnakenburgand GM models in § §
4, respectively.In § § § § R b ( kkk ) of the Bloch Green’s function for the Laplacianthat arises in the objective function characterizing the optimum lattice. Similar Green’s functions, but for theHelmholtz operator, arise in the linearized theory of the scattering of water waves by a periodic arrangement ofobstacles, and in related wave phenomena in electromagnetics and photonics. The numerical computation of BlochGreen’s functions is well-known to be a challenging problem owing to the very slow convergence of their infiniteseries representations in the spatial domain, and methodologies to improve the convergence properties based on thePoisson summation formula are surveyed in [ ] and [ ]. The numerical approach we use to compute R b ( kkk ) is anEwald summation method, based on the Poisson summation formula involving the direct and reciprocal lattices, andfollows closely the methodology developed in [ ] and [ ]. Our numerical results show that within the class of obliqueBravais lattices having a common area | Ω | of the primitive cell, it is a regular hexagonal lattice that optimizes thestability threshold for the Schnakneburg, GM, and GS models.Finally, we remark that optimal lattice arrangements of localized structures in other PDE models having a varia- D. Iron, J. Rumsey, M. J. Ward, J. Wei tional structure, such as the study of vortices in Ginzburg-Landau theory (cf. [ ]), the analysis of Abrikosov vortexlattices in the magnetic Ginzburg-Landau system (cf. [ , ]) and the study of droplets in diblock copolymer theory(cf. [ ]), have been identified through the minimization of certain energy functionals. In contrast, for our RD systemshaving no variational structure, the optimal lattice is identified not through an energy minimization criterion, butinstead from a detailed analysis that determines the spectrum of the linearization near the origin in the spectralplane when D is near a critical value. In this section we recall some basic facts about lattices and we introduce the Bloch-periodic Green’s functions thatplays a central role in the analysis in § Let lll and lll be two linearly independent vectors in R , with angle θ between them, where without loss of generalitywe take lll to be aligned with the positive x -axis. The Bravais lattice Λ is defined by(2.1) Λ = n mlll + nlll (cid:12)(cid:12)(cid:12) m, n ∈ Z o , where Z denotes the set of integers. The primitive cell is the parallelogram generated by the vectors lll and lll of area | lll × lll | . We will set the area of the primitive cell to unity, so that | lll || lll | sin θ = 1.We can also write lll , lll ∈ R as complex numbers α, β ∈ C . Without loss of generality we set Im( β ) >
0, Im( α ) = 0,and Re( α ) >
0. In terms of α and β , the area of the primitive cell is Im( α β ), which we set to unity. For a regularhexagonal lattice, | α | = | β | , with β = α e iθ , θ = π/
3, and α >
0. This yields Im( β ) = α √ / α √ / α = (4 / / . For the square lattice, we have α = 1, β = i , and θ = π/ lll , lll ∈ R , we have that lll = (cid:0) Re( α ) , Im( α ) (cid:1) , lll = (cid:0) Re( β ) , Im( β ) (cid:1) generate the lattice (2.1). For aregular hexagonal lattice of unit area for the primitive cell we have(2.2) lll = (cid:18) (cid:19) / , ! and lll = (cid:18) (cid:19) / , √ ! . In Fig. 1 we plot a portion of the hexagonal lattice generated with this lll , lll pair.The Wigner-Seitz or Voronoi cell centered at a given lattice point of Λ consists of all points in the plane that arecloser to this point than to any other lattice point. It is constructed by first joining the lattice point by a straightline to each of the neighbouring lattice points. Then, by taking the perpendicular bisector to each of these lines, theWigner-Seitz cell is the smallest area around this lattice point that is enclosed by all the perpendicular bisectors. TheWigner-Seitz cell is a convex polygon with the same area | lll × lll | of the primitive cell P . In addition, it is well-knownthat the union of the Wigner-Seitz cells for an arbitrary oblique Bravais lattice with arbitrary lattice vectors lll , lll ,and angle θ , tile all of R (cf. [ ]). In other words, there holds(2.3) R = [ z ∈ Λ ( z + Ω) . By periodicity and the property (2.3), we need only consider the Wigner-Seitz cell centered at the origin, which wedenote by Ω. In Fig. 1 we show the fundamental Wigner-Seitz cell for the hexagonal lattice. In Fig. 2 we plot theunion of the Wigner-Seitz cells for an oblique Bravais lattice with lll = (1 , lll = (cot θ,
1) and θ = 74 ◦ . he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems P S f r ag r e p l a ce m e n t s lll lll lll lll lll lll − lll − lll lll + lll lll + lll − lll + lll − lll + lll lll lll − lll − lll − lll + lll − lll + lll lll − lll lll − lll − lll − lll − lll − lll lll + lll lll + lll lll lll lll − lll lll − lll lll − lll lll − lll Figure 1. Hexagonal lattice generated by the lattice vectors (2.2). The fundamental Wigner-Seitz cell Ω for thislattice is the regular hexagon centered at the origin. The area Ω and the primitive cell are the same, and are set tounity.Figure 2. Wigner-Seitz cells for an oblique lattice with lll = (1 , lll = (cot θ, θ = 74 ◦ , so that | Ω | = 1. Thesecells tile the plane. The boundary of the Wigner-Seitz cells consist of three pairs of parallel lines of equal length.As in [ ], we define the reciprocal lattice Λ ⋆ in terms of the two independent vectors ddd and ddd , which are obtainedfrom the lattice Λ by requiring that(2.4) ddd i · lll j = δ ij , where δ ij is the Kronecker symbol. The reciprocal lattice Λ ⋆ is defined by(2.5) Λ ⋆ = n mddd + nddd (cid:12)(cid:12)(cid:12) m, n ∈ Z o . D. Iron, J. Rumsey, M. J. Ward, J. Wei
The first Brillouin zone, labeled by Ω B , is defined as the Wigner-Seitz cell centered at the origin in the reciprocalspace.We remark that other authors (cf. [ ], [ ]) define the reciprocal lattice as Λ ⋆ = { πm ddd , πn ddd } m,n ∈ Z . Ourchoice (2.5) for Λ ⋆ is motivated by the form of the Poisson summation formula of [ ] given in (6.4) below, and whichis used in § P S f r ag r e p l a ce m e n t s lll lll lll − lll lll − lll lll − lll lll − lll lll − lll lll − lll lll + lll lll − lll − lll + lll − lll − lll − lll − lll − lll − lll lll + lll lll − lll − lll − lll − lll − lll − lll + lll − lll − lll − lll − lll − lll − lll − lll − lll − lll − lll (a) Lattice Λ P S f r ag r e p l a ce m e n t s lll lll lll − lll lll − lll lll − lll lll − lll lll − lll lll − lll lll + lll lll − lll − lll + lll − lll − lll − lll − lll − lll − lll lll + lll lll − lll − lll − lll − lll − lll − lll + lll − lll − lll − lll − lll − lll − lll − lll − lll − lll − lll ddd ddd ddd − ddd ddd − ddd ddd − ddd ddd − ddd ddd − ddd ddd − ddd ddd + ddd ddd +2 ddd ddd +3 ddd ddd − ddd ddd − ddd ddd − ddd − ddd + ddd − ddd +2 ddd − ddd − ddd − ddd − ddd − ddd − ddd ddd + ddd ddd − ddd ddd − ddd − ddd − ddd − ddd +2 ddd − ddd +3 ddd − ddd − ddd − ddd +2 ddd − ddd + ddd − ddd − ddd ddd − ddd ddd − ddd ddd − ddd − ddd − ddd (b) Reciprocal Lattice Λ ∗ Figure 3. Left panel: Triangular lattice Λ with unit area of the primitive cell generated by the lattice vectors in (2.6).Right panel: the corresponding reciprocal lattice Λ ∗ with reciprocal lattice vectors as in (2.9).Finally we make some remarks on the equilateral triangular lattice which does not fall into the framework discussedabove. As observed in [ ], this special lattice requires a different treatment. For the equilateral triangle lattice, θ = 2 π/ (cid:0) e iπ/ (cid:1) = √ /
2, so that the unit area requirement of the primitive cell again yields α = (4 / / .Since Re (cid:0) e iπ/ (cid:1) = − /
2, it follows that in terms of lll i ∈ R for i = 1 ,
2, an equilateral triangle cell structure has(2.6) lll = (cid:18) (cid:19) / , ! and lll = (cid:18) (cid:19) / − , √ ! . This triangular lattice is shown in Fig. 3. The centers of the triangular cells are generated by (2.1), but there arepoints in Λ which are not cell centers (see Fig. 3). For example, (3 n +1) lll + lll , (3 n +2) lll , 3 nlll − lll , and (3 n +1) lll − lll are not centers of cells of equilateral triangles. In general, for integers p and q the point p lll + qlll will be a vertexinstead of a cell center when(2.7) ( p mod 3) + ( q mod 3) = 2 , where the positive representation of the mod function is used, i.e. ( −
1) mod 3 = 2 . Thus, for the equilateral triangularlattice the set of lattice points is(2.8) Λ tri = n mlll + nlll (cid:12)(cid:12)(cid:12) m, n ∈ Z , ( m mod 3) + ( n mod 3) = 2 o . he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems lll and lll given by (2.6), the definingvectors for Λ ⋆ are(2.9) ddd = 112 / (cid:16) √ , (cid:17) and ddd = 112 / (0 , , as can be verified by substitution into (2.4). A plot of a portion of this reciprocal lattice for the equilateral trianglelattice is shown in the right panel of Fig. 3. From this plot it follows that, for integer p and q , p ddd + q ddd will be avertex, not a centre, when(2.10) ( p − q ) mod 3 = 1 . Therefore the reduced reciprocal lattice becomes(2.11) Λ ⋆tri = n mddd + nddd (cid:12)(cid:12)(cid:12) m, n ∈ Z , ( m − n ) mod 3 = 1 o . Unfortunately for the equilateral triangular lattice the property (2.3) does not hold. In other words, the whole R is not the union of cells translated on the Bravais lattice, and thus one can not restrict to one Wigner-Seitz cell atthe origin. As such, it is unclear whether the corresponding Poisson summation formula in (6.4) below still holds.However, if a homogeneous Neumann boundary condition is imposed on the cell, it is possible to reflect through theedges and fill the whole R . (This fact has been used in [ ].) Therefore, the equilibrium contruction of a periodicspot pattern presented in Section 3.1 and Section 4.1 still applies for the equilateral triangular lattice. However, thestability of periodic spot patterns on the triangular lattice is an open problem. In our analysis of the stability of spot patterns in § § G b ( x ) for theLaplacian plays a prominent role. In the Wigner-Seitz cell Ω, G b ( x ) for kkk/ (2 π ) ∈ Ω B , satisfies(2.12 a ) ∆ G b = − δ ( x ) ; x ∈ Ω , subject to the quasi-periodicity condition on R that(2.12 b ) G b ( x + lll ) = e − ikkk · lll G b ( x ) , lll ∈ Λ , where Λ is the Bravais lattice (2.1). As we show below, (2.12 b ) indirectly yields boundary conditions on the boundary ∂ Ω of the Wigner-Seitz cell. The regular part R b ( kkk ) of this Bloch Green’s function is defined by(2.12 c ) R b ( kkk ) ≡ lim x → (cid:18) G b ( x ) + 12 π log | x | (cid:19) . In order to study the properties of G b ( x ) and R b ( kkk ), we first require a more refined description of the Wigner-Seitzcell. To do so, we observe that there are eight nearest neighbor lattice points to x = 0 given by the set(2.13) P ≡ { mlll + nlll | m ∈ { , , − } , n ∈ { , , − } , ( m, n ) = 0 } . For each (vector) point
PPP i ∈ P , for i = 1 , . . . ,
8, we define a Bragg line L i . This is the line that crosses the point PPP i / PPP i . We define the unit outer normal to L i by ηηη i ≡ PPP i / | PPP i | . The convex hull generated by theseBragg lines is the Wigner-Seitz cell Ω, and the boundary ∂ Ω of the Wigner-Seitz cell is, generically, the union of sixBragg lines. For a square lattice, ∂ Ω has four Bragg lines. The centers of the Bragg lines generating ∂ Ω are re-indexed
D. Iron, J. Rumsey, M. J. Ward, J. Wei as PPP i for i = 1 , . . . , L , where L ∈ { , } is the number of Bragg lines de-marking ∂ Ω. The boundary ∂ Ω of Ω is theunion of the re-indexed Bragg lines L i , for i = 1 , . . . , L , and is parametrized segment-wise by a parameter t as(2.14) ∂ Ω = n x ∈ [ i { PPP i tηηη ⊥ i } (cid:12)(cid:12)(cid:12) − t i ≤ t ≤ t i , i = 1 , . . . , L , L = { , } o . Here 2 t i is the length of L i , and ηηη ⊥ i is the direction perpendicular to PPP i , and therefore tangent to L i .The following observation is central to the analysis below: Suppose that PPP is a neighbor of 0 and that the Braggline crossing
PPP / ∂ Ω. Then, by symmetry, the Bragg line crossing − PPP / ∂ Ω. In other words,Bragg lines on ∂ Ω must come in pairs. This fact is evident from the plot of the Wigner-Seitz cell for the obliquelattice shown in Fig. 2. With this more refined description of the Wigner-Seitz cell, we now state and prove two keyLemmas that are needed in § § Lemma 2.1
The regular part R b ( kkk ) of the Bloch Green’s function G b ( x ) satisfying (2.12) is real-valued for | kkk | 6 = 0 . Proof:
Let 0 < ρ ≪ ρ ≡ Ω − B ρ (0), where B ρ (0) is the ball of radius ρ centered at x = 0. We multiply(2.12 a ) by ¯ G b , where the bar denotes conjugation, and we integrate over Ω ρ using the divergence theorem to get(2.15) Z Ω ρ ¯ G b ∆ G b d x + Z Ω ρ ∇ ¯ G b · ∇ G b d x = Z ∂ Ω ρ ¯ G b ∂ ν G b d x = Z ∂ Ω ¯ G b ∂ ν G b d x − Z ∂B ρ ( ) ¯ G b ∂ | x | G b d x . Here ∂ ν G b denotes the outward normal derivative of G b on ∂ Ω. For ρ ≪
1, we use (2.12 c ) to calculate(2.16) Z ∂B ρ ( ) ¯ G b ∂ | x | G b d x ∼ π Z (cid:18) − π log ρ + R b ( kkk ) + o (1) (cid:19) (cid:18) − πρ + O (1) (cid:19) ρ dθ ∼ π log ρ − R b ( kkk ) + O ( ρ log ρ ) . Upon using (2.16), together with ∆ G b = 0 in Ω ρ , in equation (2.15), we let ρ → R b ( kkk ) = − Z ∂ Ω ¯ G b ( x ) ∂ ν G b ( x ) d x + lim ρ → h Z Ω ρ |∇ G b | d x + 12 π log ρ i . From (2.17), to show that R b ( kkk ) is real-valued it suffices to establish that the boundary integral term in (2.17)vanishes. To show this, we observe that since the Bragg lines come in pairs, we have(2.18) Z ∂ Ω ¯ G b ( x ) ∂ ν G b ( x ) d x = L/ X i =1 Z PPPi + tηηη ⊥ i ¯ G b ( x ) ∇ x G b ( x ) · ηηη i d x − Z − PPPi + tηηη ⊥ i ¯ G b ( x ) ∇ x G b ( x ) · ηηη i d x . Here we have used the fact that the outward normals to the Bragg line pairs
PPP i / tηηη ⊥ i and − PPP i / tηηη ⊥ i are inopposite directions. We then translate x by PPP i to get(2.19) Z PPPi + tηηη ⊥ i ¯ G b ( x ) ∇ x G b ( x ) · ηηη i d x = Z − PPPi + tηηη ⊥ i + PPP i ¯ G b ( x ) ∇ x G b ( x ) · ηηη i d x = Z − PPPi + tηηη ⊥ i ¯ G b ( x + PPP i ) ∇ x G b ( x + PPP i ) · ηηη i d x . Then, since
PPP i ∈ Λ, we have by the quasi-periodicity condition (2.12 b ) that¯ G b ( x + PPP i ) ∇ x G b ( x + PPP i ) = (cid:16) ¯ G b ( x ) e ikkk · PPP i (cid:17) (cid:16) ∇ x G b ( x ) e − ikkk · PPP i (cid:17) = ¯ G b ( x ) ∇ x G b ( x ) . he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems Z PPPi + tηηη ⊥ i ¯ G b ( x ) ∇ x G b ( x ) · ηηη i d x = Z − PPPi + tηηη ⊥ i ¯ G b ( x ) ∇ x G b ( x ) · ηηη i d x , which establishes from (2.18) that R ∂ Ω ¯ G b ( x ) ∂ ν G b ( x ) d x = 0. From (2.17) we conclude that R b ( kkk ) is real. (cid:4) Next, we determine the asymptotic behavior of R b ( kkk ) as | kkk | →
0. Since (2.12) has no solution if kkk = 0, it suggeststhat R b ( kkk ) is singular as | kkk | →
0. To determine the asymptotic behavior of G b as | kkk | →
0, we introduce a smallparameter σ ≪
1, and define kkk = σκκκ where | κκκ | = O (1). For σ ≪
1, we expand G b ( x ) as(2.20) G b ( x ) = σ − U ( x ) + σ − U ( x ) + U ( x ) + · · · . For any lll ∈ Ω, and for σ ≪
1, we have from (2.12 b ) that(2.21) U ( x + lll ) σ + U ( x + lll ) σ + U ( x + lll ) + · · · = (cid:20) − iσ ( κκκ · lll ) − σ κκκ · lll ) + · · · (cid:21) (cid:18) U ( x ) σ + U ( x ) σ + U ( x ) + · · · (cid:19) . Upon substituting (2.20) into (2.12 a ), and then equating powers of σ in (2.21), we obtain the sequence of problems∆ U = 0 ; U ( x + lll ) = U ( x ) , (2.22 a ) ∆ U = 0 ; U ( x + lll ) = U ( x ) − i ( κκκ · lll ) U ( x ) , (2.22 b ) ∆ U = − δ ( x ) ; U ( x + lll ) = U ( x ) − i ( κκκ · lll ) U ( x ) − ( κκκ · lll ) U ( x ) . (2.22 c )The solution to (2.22 a ) is that U is an arbitrary constant, while the solution to (2.22 b ) is readily calculated as U ( x ) = − i ( κκκ · x ) U + U , where U is an arbitrary constant. Upon substituting U and U into (2.22 c ), we obtainfor any lll ∈ Λ that U satisfies(2.23) ∆ U = − δ ( x ) ; U ( x + lll ) = U ( x ) − ( κκκ · lll ) ( κκκ · x ) U − i ( κκκ · lll ) U − ( κκκ · lll ) U . By differentiating the periodicity condition in (2.23) with respect to x , we have for any lll ∈ Λ that(2.24) ∇ x U ( x + lll ) = ∇ x U ( x ) − κκκ ( κκκ · lll ) U . Next, to determine U , we integrate ∆ U = 0 over Ω to obtain from the divergence theorem and a subsequentdecomposition of the boundary integral over the Bragg line pairs, as in (2.18), that(2.25) − Z ∂ Ω ∂ ν U d x = L/ X i =1 Z PPPi + tηηη ⊥ i ∇ x U ( x ) · ηηη i d x − Z − PPPi + tηηη ⊥ i ∇ x U ( x ) · ηηη i d x . Then, as similar to the derivation in (2.19), we calculate the boundary integrals as(2.26) Z PPPi + tηηη ⊥ i ∇ x U ( x ) · ηηη i d x = Z − PPPi + tηηη ⊥ i + PPP i ∇ x U ( x ) · ηηη i d x = Z − PPPi + tηηη ⊥ i ∇ x U ( x + PPP i ) · ηηη i d x . Upon using (2.26) in (2.25), we obtain(2.27) − L/ X i =1 Z − PPPi + tηηη ⊥ i ( ∇ x U ( x + PPP i ) − ∇ x U ( x )) · ηηη i d x . D. Iron, J. Rumsey, M. J. Ward, J. Wei
Since
PPP i ∈ Λ and ηηη i = PPP i / | PPP i | , we calculate the integrand in (2.27) by using (2.24) as(2.28) ( ∇ x U ( x + PPP i ) − ∇ x U ( x )) · ηηη i = − ( κκκ · ηηη i ) ( κκκ · PPP i ) U = − ( κκκ · PPP i ) U | PPP i | . Then, upon substituting (2.28) into (2.27), and by integrating the constant integrand over the Bragg lines, weobtain that U satisfies(2.29) − −U L/ X i =1 ( κκκ · PPP i ) | PPP i | t i = −U L X i =1 ( κκκ · PPP i ) | PPP i | t i = −U L X i =1 ( κκκ · ηηη i ) t i | PPP i | , where 2 t i is the length of the Bragg line L i . Upon solving for U , we obtain that(2.30) U = 1 κκκ T Q κκκ , where Q ≡ L X i =1 ηηη i ω i ηηη Ti , and ω i ≡ t i | PPP i | . Since ω i >
0, for i = 1 , . . . , L , we have y T Q y = P Li =1 (cid:0) ηηη Ti y (cid:1) ω i > y = 0, which proves that the matrix Q is positive definite. We summarize the results of this perturbation calculation in the following (formal) lemma: Lemma 2.2
For | kkk | → , the regular part R b ( kkk ) of the Bloch Green’s function of (2.12) has the leading-ordersingular asymptotic behavior (2.31) R b ( kkk ) ∼ kkk T Q kkk , where the positive definite matrix Q is defined in terms of the parameters of the Wigner-Seitz cell by (2.30). We remark that a similar analysis can be done for the quasi-periodic reduced-wave Green’s function, which satisfies(2.32 a ) ∆ G ( x ) − σ G = − δ ( x ) ; x ∈ Ω ; G ( x + lll ) = e − ikkk · lll G ( x ) , lll ∈ Λ , where Λ is the Bravais lattice (2.1) and kkk/ (2 π ) ∈ Ω B . The regular part R ( kkk ) of this Green’s function is defined by(2.32 b ) R ( kkk ) ≡ lim x → (cid:18) G ( x ) + 12 π log | x | (cid:19) . By a simple modification of the derivation of Lemma 2.1 and 2.2, we obtain the following result:
Lemma 2.3
Let kkk/ (2 π ) ∈ Ω B . For the regular part R ( kkk ) of the reduced-wave Bloch Green’s function satisfying(2.32), we have the following: • (i) Let σ be real. Then R ( kkk ) is real-valued. • (ii) R ( kkk ) ∼ R b ( kkk ) + O ( σ ) for σ → when | kkk | > with | kkk | = O (1) . Here R b ( kkk ) is the regular part of the BlochGreen’s function (2.12). • (iii) Let σ → , and consider the long-wavelength regime | kkk | = O ( σ ) , where kkk = σκκκ with | κκκ | = O (1) . Then, (2.33) R ( kkk ) ∼ σ [ | Ω | + κκκ T Q κκκ ] , where the positive definite matrix Q is defined in (2.30).he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems Proof:
To prove (i) we proceed as in the derivation of Lemma 2.1 to get(2.34) R ( kkk ) = lim ρ → h Z Ω ρ (cid:0) |∇ G | + σ | G | (cid:1) d x + 12 π log ρ i , which is real-valued. The second result (ii) is simply a regular perturbation result for the solution to (2.32) for σ → | kkk | is bounded away from zero and kkk/ (2 π ) ∈ Ω B , so that kkk · lll = 2 πN . Therefore, when kkk/ (2 π ) ∈ Ω B , R ( kkk ) isunbounded only as | kkk | → U = U − δ ( x ) in Ω, Therefore, we must add the term U | Ω | to the left-hand sides of (2.25), (2.27), and (2.29). Bysolving for U we get (2.33). (cid:4) In § § ε →
0, it is the eigenfunction Ψ corresponding to the long-rangesolution component u that satisfies an elliptic PDE with coefficients that are spatially periodic on the lattice. Assuch, by the Floquet-Bloch theorem (cf. [ ] and [ ]), this eigenfunction must satisfy the quasi-periodic boundaryconditions Ψ( x + lll ) = e − ikkk · lll Ψ( x ) for lll ∈ Λ, x ∈ R and kkk/ (2 π ) ∈ Ω B . This quasi-periodicity condition can be usedto formulate a boundary operator on the boundary ∂ Ω of the fundamental Wigner-Seitz cell Ω. Let L i and L − i betwo parallel Bragg lines on opposite sides of ∂ Ω for i = 1 , . . . , L/
2. Let x i ∈ L i and x i ∈ L − i be any two opposingpoints on these Bragg lines. We define the boundary operator P k Ψ by(2.35) P k Ψ = n Ψ (cid:12)(cid:12)(cid:12) (cid:18) Ψ( x i ) ∂ n Ψ( x i ) (cid:19) = e − ikkk · lll i (cid:18) Ψ( x i ) ∂ n Ψ( x i ) (cid:19) , ∀ x i ∈ L i , ∀ x i ∈ L − i , lll i ∈ Λ , i = 1 , . . . , L/ o . The boundary operator P Ψ simply corresponds to a periodicity condition for Ψ on each pair of parallel Bragg lines.These boundary operators are used in § § We study the linear stability of a steady-state periodic pattern of localized spots for the Schnakenburg model (1.1)where the spots are centered at the lattice points of (2.1). The analysis below is based on the fundamental Wigner-Seitz cell Ω, which contains exactly one spot centered at the origin.
We use the method of matched asymptotic expansions to construct a steady-state one-spot solution to (1.1) centeredat x = 0 ∈ Ω. The construction of such a solution consists of an outer region where v is exponentially small and u = O (1), and an inner region of extent O ( ε ) centered at the origin where both v and u have localized.In the inner region we look for a locally radially symmetric steady-state solution of the form(3.1) u = 1 √ D U , v = √ DV , y = ε − x . Then, substituting (3.1) into the steady-state equations of (1.1), we obtain that V ∼ V ( ρ ) and U ∼ U ( ρ ), with ρ = | y | ,satisfy the following core problem in terms of an unknown source strength S ≡ R ∞ U V ρ dρ to be determined:∆ ρ V − V + U V = 0 , ∆ ρ U − U V = 0 , < ρ < ∞ , (3.2 a ) U ′ (0) = V ′ (0) = 0 ; V → , U ∼ S log ρ + χ ( S ) + o (1) , as ρ → ∞ . (3.2 b )Here we have defined ∆ ρ V ≡ V ′′ + ρ − V ′ .2 D. Iron, J. Rumsey, M. J. Ward, J. Wei
The core problem (3.2), without the explicit far-field condition (3.2 b ) was first identified and its solutions computednumerically in § ]. In [ ], the function χ ( S ) was computed numerically, and solutions to the core problemwere shown to closely related to the phenomena of self-replicating spots.The unknown source strength S is determined by matching the the far-field behavior of the core solution to anouter solution for u valid away from O ( ε ) distances from 0. In the outer region, v is exponentially small, and from(3.1) we get ε − uv → π √ DSδ ( x ). Therefore, from (1.1), the outer steady-state problem for u is∆ u = − aD + 2 π √ D S δ ( x ) , x ∈ Ω ; P u = 0 , x ∈ ∂ Ω ,u ∼ √ D (cid:20) S log | x | + χ ( S ) + Sν (cid:21) , as x → , (3.3)where ν ≡ − / log ε and Ω is the fundamental Wigner-Seitz cell. The divergence theorem then yields(3.4) S = a | Ω | π √ D .
The solution to (3.3) is then written in terms of the periodic Green’s function G p ( x ) as(3.5) u ( x ) = − π √ D [ SG p ( x ; 0) − u c ] , u c ≡ πν [ S + 2 πνSR p + νχ ( S )] , where the periodic source-neutral Green’s function G p ( x ) and its regular part R p satisfy∆ G p = 1 | Ω | − δ ( x ) , x ∈ Ω ; P G p = 0 x ∈ ∂ Ω ,G p ∼ − π log | x | + R p + o (1) , as x → Z Ω G p d x = 0 . (3.6)An explicit expression for R p on an oblique Bravais lattice was derived in Theorem 1 of [ ]. A periodic pattern ofspots is then obtained through periodic extension to R of the one-spot solution constructed within Ω.Since the stability threshold occurs when D = O (1 /ν ), for which S = O ( ν / ) ≪ ν for the solution to the core problem (3.2). This result, which is required forthe stability analysis in § Lemma 3.1
For S = S ν / + S ν / + · · · , where ν ≡ − / log ε ≪ , the asymptotic solution to the core problem(3.2) is (3.7 a ) V ∼ ν / ( V + νV + · · · ) , U ∼ ν − / (cid:0) U + νU + ν U + · · · (cid:1) , χ ∼ ν − / ( χ + νχ + · · · ) , where U , U ( ρ ) , V ( ρ ) , and V ( ρ ) are defined by (3.7 b ) U = χ , U = χ + 1 χ U p , V = wχ , V = − χ χ w + 1 χ V p . Here w ( ρ ) is the unique ground-state solution to ∆ ρ w − w + w = 0 with w (0) > , w ′ (0) = 0 , and w → as ρ → ∞ .In terms of w ( ρ ) , the functions U p and V p are the unique solutions on ≤ ρ < ∞ to L V p = − w U p , V ′ p (0) = 0 , V p → , as ρ → ∞ , ∆ ρ U p = w , U ′ p (0) = 0 , U p → b log ρ + o (1) , as ρ → ∞ ; b ≡ ∞ Z w ρ dρ , (3.7 c ) where the linear operator L is defined by L V p ≡ ∆ ρ V p − V p + 2 wV p . Finally, in (3.7 a), the constants χ andhe Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems χ are related to S and S by (3.7 d ) χ = bS , χ = − S bS + S b ∞ Z V p ρ dρ . The derivation of this result was given in § ] and is outlined in Appendix A below. We remark that the o (1)condition in the far-field behavior of U p in (3.7 c ) eliminates an otherwise arbitrary constant in the determinationof U p . This condition, therefore, ensures that the solution to the linear BVP system (3.7 c ) is unique. To study the stability of the periodic pattern of spots with respect to fast O (1) time-scale instabilities, we usethe Floquet-Bloch theorem that allows us to only consider the Wigner-Seitz cell Ω, centered at the origin, withquasi-periodic boundary conditions imposed on its boundaries.We linearize around the steady-state u e and v e , as calculated in § u = u e + e λt η , v = v e + e λt φ . By substituting (3.8) into (1.1) and linearizing, we obtain the following eigenvalue problem for φ and η : ε ∆ φ − φ + 2 u e v e φ + v e η = λφ , x ∈ Ω ; P kkk φ = 0 , x ∈ ∂ Ω ,D ∆ η − ε − u e v e φ − ε − v e η = λτ η , x ∈ Ω ; P kkk η = 0 , x ∈ ∂ Ω , (3.9)where P kkk is the quasi-periodic boundary operator of (2.35).In the inner region near x = 0 we introduce the local variables N ( ρ ) and Φ( ρ ) by(3.10) η = 1 D N ( ρ ) , φ = Φ( ρ ) , ρ = | y | , y = ε − x . Upon substituting (3.10) into (3.9), and by using u e ∼ U ( ρ ) / √ D and v e ∼ √ DV ( ρ ), where U and V satisfy the coreproblem (3.2), we obtain on 0 < ρ < ∞ that∆ ρ Φ − Φ+2
U V
Φ +
N V = λ Φ , Φ → , as ρ → ∞ , ∆ ρ N = 2 U V
Φ +
N V , N ∼ C log ρ + B , as ρ → ∞ , (3.11)with Φ ′ (0) = N ′ (0) = 0, and where B = B ( S ; λ ). We remark that for Re( λ + 1) >
0, Φ in (3.11) decays exponentiallyas ρ → ∞ . However, in contrast, we cannot apriori impose that N in (3.11) is bounded as ρ → ∞ . Instead we mustallow for the possibility of a logarithmic growth for N as ρ → ∞ . Upon using the divergence theorem we identify C as C = R ∞ (cid:0) U V
Φ +
N V (cid:1) ρ dρ . The constant C will be determined by matching N to an outer eigenfunction η ,valid away from x = 0, that satisfies (3.9).To formulate this outer problem, we obtain since v e is localized near x = 0 that, in the sense of distributions,(3.12) ε − (cid:0) u e v e φ + ηv e (cid:1) → Z R (cid:0) U V
Φ +
N V (cid:1) d y δ ( x ) = 2 πCδ ( x ) . By using this expression in (3.9), we conclude that the outer problem for η is∆ η − τ λD η = 2 πCD δ ( x ) , x ∈ Ω ; P kkk η = 0 , x ∈ ∂ Ω ,η ∼ D (cid:18) C log | x | + Cν + B (cid:19) , as x → . (3.13)4 D. Iron, J. Rumsey, M. J. Ward, J. Wei
The solution to (3.13) is η = − πCD − G bλ ( x ), where G bλ satisfies∆ G bλ − τ λD G bλ = − δ ( x ) , x ∈ Ω ; P kkk G bλ = 0 , x ∈ ∂ Ω ,G bλ ∼ − π log | x | + R bλ , as x → . (3.14)From the requirement that the behavior of η as x → B + C/ν = − πCR bλ .Finally, since the stability threshold occurs in the regime D = O ( ν − ) ≫
1, we conclude from Lemma 2.3 (ii) thatfor | kkk | 6 = 0 and kkk/ (2 π ) ∈ Ω B ,(3.15) (cid:0) πνR b + O ( ν ) (cid:1) C = − νB , where R b is the regular part of the Bloch Green’s function G b defined by (2.12) on Ω.We now proceed to determine the portion of the continuous spectrum of the linearization that lies within an O ( ν )neighborhood of the origin, i. e. that satisfies | λ | ≤ O ( ν ), when D is close to a certain critical value. To do so, wefirst must calculate an asymptotic expansion for the solution to (3.11) together with (3.15).By using (3.7 a ) we first calculate the coefficients in the differential operator in (3.11) as U V = w + ν ( U V + U V ) + · · · = w + νχ [ V p + wU p ] + · · · ,V = ν (cid:0) V + 2 νV V (cid:1) + · · · = ν w χ + 2 ν χ (cid:18) − χ w + wV p χ (cid:19) + · · · , so that the local problem (3.11) on 0 < ρ < ∞ becomes∆ ρ Φ − Φ + (cid:20) w + 2 νχ ( V p + wU p ) + · · · (cid:21) Φ = − ν (cid:20) w χ + 2 νχ (cid:18) − χ w + wV p χ (cid:19) + · · · (cid:21) N + λ Φ , ∆ ρ N = (cid:20) w + 2 νχ ( V p + wU p ) + · · · (cid:21) Φ + ν (cid:20) w χ + 2 νχ (cid:18) − χ w + wV p χ (cid:19) + · · · (cid:21) N , Φ → , N ∼ C log ρ + B , as ρ → ∞ ; Φ ′ (0) = N ′ (0) = 0 . (3.16)We then introduce the appropriate expansions N = 1 ν (cid:16) ˆ N + ν ˆ N + · · · (cid:17) , B = 1 ν (cid:16) ˆ B + ν ˆ B + · · · (cid:17) , C = C + νC + · · · , Φ = Φ + ν Φ + · · · , λ = λ + νλ + · · · , (3.17)into (3.16) and collect powers of ν .To leading order, we obtain on 0 < ρ < ∞ that L Φ ≡ ∆ ρ Φ − Φ + 2 w Φ = − w χ ˆ N + λ Φ , ∆ ρ ˆ N = 0 , Φ → , ˆ N → ˆ B as ρ → ∞ ; Φ ′ (0) = 0 , ˆ N ′ (0) = 0 , (3.18)where L is referred to as the local operator. We conclude that ˆ N = ˆ B for ρ ≥ ρ > satisfies(3.19) L Φ + 2 χ ( V p + wU p ) Φ + 2 χ (cid:18) − χ w + wV p χ (cid:19) ˆ N = − w χ ˆ N + λ Φ ; Φ → , as ρ → ∞ , with Φ ′ (0) = 0, and that ˆ N on ρ > ρ ˆ N = 2 w Φ + w χ ˆ N ; ˆ N ∼ C log ρ + ˆ B , as ρ → ∞ ; ˆ N ′ (0) = 0 . he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems N given by∆ ρ ˆ N = 2 w Φ + 2 χ ( V p + wU p ) Φ + 2 χ (cid:18) − χ w + wV p χ (cid:19) ˆ N + w χ ˆ N , ˆ N ∼ C log ρ + ˆ B , as ρ → ∞ ; ˆ N ′ (0) = 0 . (3.21)In addition, by substituting (3.17) into (3.15), we obtain upon collecting powers of ν that(3.22) C = − ˆ B , C + 2 πR b C = − ˆ B . Next, we proceed to analyze (3.18)–(3.21). From the divergence theorem, we obtain from (3.20) that(3.23) C = ∞ Z w Φ ρ dρ + bχ ˆ N , b ≡ ∞ Z w ρ dρ . Since C = − ˆ B and ˆ B = ˆ N , (3.23) yields that(3.24) ˆ N = ˆ B = − (cid:20) bχ (cid:21) − ∞ Z w Φ ρ dρ . With ˆ N known, (3.18) provides the leading-order nonlocal eigenvalue problem (NLEP)(3.25) L Φ − w bχ + b R ∞ w Φ ρ dρ R ∞ w ρ dρ = λ Φ ; Φ → , as ρ → ∞ ; Φ ′ (0) = 0 . For this NLEP, the rigorous result of [ ] (see also Theorem 3.7 of the survey article [ ]) proves that Re( λ ) < b/ ( χ + b ) >
1. At the stability threshold where 2 b/ ( χ + b ) = 1, we have from the identity L w = w that Φ = w and λ = 0. From (3.24) and (3.23) we can then calculate ˆ B and C at this leading-order stabilitythreshold. In summary, to leading order in ν , we obtain at λ = 0 that(3.26) bχ = 1 , Φ = w , ˆ B = ˆ N = − b = − ∞ Z w ρ dρ , C = b . Upon substituting (3.26) into (3.20) we obtain at λ = 0 that ˆ N on ρ > ρ ˆ N = w ; ˆ N ∼ b log ρ + ˆ B , as ρ → ∞ ; ˆ N ′ (0) = 0 . Upon comparing (3.27) with the problem for U p , as given in (3.7 c ), we conclude that(3.28) ˆ N = U p + ˆ B . Next, we observe that for D = D /ν ≫
1, it follows from (3.4) that S = ν / S + · · · , where S = a | Ω | / (2 π √ D ).Then, since S = b/χ from (3.7 d ), and b/χ = 1 when λ = 0 from (3.26), the critical value of D at the leading-orderstability threshold λ = 0 is(3.29 a ) D = D c ≡ a | Ω | π b . This motivates the definition of the bifurcation parameter µ by(3.29 b ) µ ≡ π Dνba | Ω | , so that at criticality where χ = √ b , we have µ = 1.We then proceed to analyze the effect of the higher order terms in powers of ν on the stability threshold. In6 D. Iron, J. Rumsey, M. J. Ward, J. Wei particular, we determine the continuous band of spectrum that is contained within an O ( ν ) ball near λ = 0 whenthe bifurcation parameter µ is O ( ν ) close to its leading-order critical value µ = 1. As such, we set(3.30) λ = νλ + · · · , for µ = 1 + νµ + · · · , and we derive an expression for λ in terms of µ , the Bloch vector kkk , the lattice structure, and certain correctionterms to the core problem.To determine an expression for µ in terms of χ and χ we first set D = D /ν and write the two term expansionfor the source strength S as S = a | Ω | π √ D = ν / ( S + νS + · · · ) , where S and S are given in (3.7 d ) in terms of χ and χ . By using (3.7 d ) and (3.29 b ), this expansion for S becomes(3.31) s bµ = bχ + ν − χ bχ + 1 χ ∞ Z V p ρ dρ + · · · . As expected, to leading order we have µ = 1 when b = χ . At λ = 0 where χ = √ b , we use µ − / ∼ − νµ / · · · to relate µ to χ as(3.32) χ √ b = µ b ∞ Z V p ρ dρ . Next, we substitute Φ = w , ˆ N = − b , χ = b , and ˆ N = U p + ˆ B from (3.28), into the equation (3.19) for Φ .After some algebra, we conclude that Φ at λ = 0 satisfies(3.33) L Φ + w b ˆ B = − χ χ b w − b w U p + λ w ; Φ → , as ρ → ∞ , with Φ ′ (0) = 0. In a similar way, at the leading-order stability threshold, the problem (3.21) for ˆ N on ρ > ρ ˆ N = 2 w Φ + w b ˆ B + 3 b w U p + 2 χ χ b w , ˆ N ∼ C log ρ + ˆ B , as ρ → ∞ ; ˆ N ′ (0) = 0 . (3.34)To determine ˆ B , as required in (3.33), we use the divergence theorem on (3.34) to obtain that C = 2 ∞ Z w Φ ρ dρ + ˆ B + 3 b ∞ Z w U p ρ dρ + 2 χ χ . Upon combining this expression with C + 2 πR b C = − ˆ B , as obtained from (3.22), where C = b , we obtain ˆ B asˆ B = − ∞ Z w Φ ρ dρ − πbR b − b ∞ Z w U p ρ dρ − χ χ . Upon substituting this expression into (3.33), we conclude that Φ satisfies(3.35 a ) L Φ ≡ L Φ − w R ∞ w Φ ρ dρ R ∞ w ρ dρ = R s + λ w ; Φ → , as ρ → ∞ , with Φ ′ (0) = 0, where the residual R s is defined by(3.35 b ) R s ≡ πw R b + 32 b w ∞ Z w U p ρ dρ − χ χ w b − b w U p . he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems λ is determined by imposing a solvability condition on (3.35). The homogeneous adjoint operator L ⋆ corresponding to (3.35) is(3.36) L ⋆ Ψ ≡ L Ψ − w R ∞ w Ψ ρ dρ R ∞ w ρ dρ . We define Ψ ⋆ = w + ρw ′ / L Ψ ⋆ = w and L w = w (see [ ]). Then, we use Green’ssecond identity to obtain R ∞ [ wL Ψ ⋆ − Ψ ⋆ L w ] ρ dρ = R ∞ (cid:0) w − Ψ ⋆ w (cid:1) ρ dρ . By the decay of w and Ψ ⋆ as ρ → ∞ ,we obtain that R ∞ w ρ dρ = R ∞ Ψ ⋆ w ρ dρ . Therefore, since the ratio of the two integrals in (3.36) is unity whenΨ = Ψ ⋆ , we conclude that L ⋆ Ψ ⋆ = 0.Finally, we impose the solvability condition that the right hand side of (3.35) is orthogonal to Ψ ⋆ in the sense that λ R ∞ w Ψ ⋆ ρ dρ + R ∞ R s Ψ ∗ ρ dρ = 0. By using (3.35 b ) for R s , this solvability condition yields that(3.37) λ = − R ∞ w Ψ ⋆ ρ dρb R ∞ w Ψ ⋆ ρ dρ bπR b − χ χ + 32 b ∞ Z w U p ρ dρ − R ∞ w U p Ψ ⋆ ρ dρ R ∞ w Ψ ⋆ ρ dρ . Equation (3.37) is simplified by first calculating the following integrals by using integration by parts: ∞ Z w Ψ ⋆ ρ dρ = ∞ Z ( L w ) (cid:0) L − w (cid:1) = ∞ Z w ρ dρ = b , ∞ Z w Ψ ⋆ ρ dρ = ∞ Z ρw (cid:16) w + ρ w ′ (cid:17) dρ = ∞ Z w ρ dρ + 14 ∞ Z (cid:2) w (cid:3) ′ ρ dρ = b . (3.38)In addition, since L V p = − w U p from (3.7 c ) and Ψ ⋆ = L − w , we obtain upon integrating by parts that ∞ Z w U p Ψ ⋆ ρ dρ = − ∞ Z ( L V p ) (cid:0) L − w (cid:1) ρ dρ = − ∞ Z V p wρ dρ . By substituting this expression and (3.38) into (3.37), we obtain(3.39) λ − b bπR b − χ χ + 32 b ∞ Z w U p ρ dρ + 3 b ∞ Z wV p ρ dρ . Next, we use (3.7 c ) to calculate R ∞ w U p ρ dρ = R ∞ ( V p − wV p ) ρ dρ . Finally, we substitute this expressiontogether with χ = √ b and (3.32), which relates µ to χ , into (3.39) to obtain our final expression for λ . Wesummarize our result as follows: Principal Result 3.1
In the limit ε → , consider a steady-state periodic pattern of spots for the Schnakenburgmodel (1.1) on the Bravais lattice Λ when D = O ( ν − ) , where ν = − / log ε . Then, when (3.40 a ) D = a | Ω | π bν (1 + µ ν ) , where µ = O (1) , the portion of the continuous spectrum of the linearization that lies within an O ( ν ) neighborhoodof the origin λ = 0 , i. e. that satisfies | λ | ≤ O ( ν ) , is given by (3.40 b ) λ = νλ + · · · , λ = 2 µ − πR b − b ∞ Z ρV p dρ . D. Iron, J. Rumsey, M. J. Ward, J. WeiHere | Ω | is the area of the Wigner-Seitz cell and R b = R b ( kkk ) is the regular part of the Bloch Green’s function G b defined on Ω by (2.12), with kkk = 0 and kkk/ (2 π ) ∈ Ω B . The result (3.40 b ) determines how the portion of the band of continuous spectrum that lies near the origin dependson the de-tuning parameter µ , the correction V p to the solution of the core problem, and the lattice structure andBloch wavevector kkk as inherited from R b ( kkk ). Remark 3.1
We need only consider kkk/ (2 π ) in the first Brillouin zone Ω B , defined as the Wigner-Seitz cell centeredat the origin for the reciprocal lattice. Since R b is real-valued from Lemma 2.1, it follows that the band of spectrum(3.40 b) lies on the real axis in the λ -plane. Furthermore, since by Lemma 2.2, R b = O (cid:16) / ( kkk T Q kkk ) (cid:17) → + ∞ as | kkk | → for some positive definite matrix Q , the continuous band of spectrum that corresponds to small values of | kkk | is not within an O ( ν ) neighborhood of λ = 0 , but instead lies at an O (cid:16) ν/kkk T Q kkk (cid:17) distance from the origin along thenegative real axis in the λ -plane. We conclude from (3.40 b ) that a periodic arrangement of spots with a given lattice structure is linearly stablewhen(3.41) µ < πR ⋆b + 1 b ∞ Z V p ρ dρ , R ⋆b ≡ min kkk R b ( kkk ) . For a fixed area | Ω | of the Wigner-Seitz cell, the optimal lattice geometry is defined as the one that allows for stabilityfor the largest inhibitor diffusivity D . This leads to one of our main results. Principal Result 3.2
The optimal lattice arrangement for a periodic pattern of spots for the Schnakenburg model(1.1) is the one for which K s ≡ R ∗ b is maximized. Consequently, this optimal lattice allows for stability for the largestpossible value of D . For ν = − / log ε ≪ , a two-term asymptotic expansion for this maximal stability threshold for D is given explicitly by in terms of an objective function K s by (3.42) D optim ∼ a | Ω | π bν ν π max Λ K s + 1 b ∞ Z V p ρ dρ , K s ≡ R ⋆b = min kkk R b , where max Λ K s is taken over all lattices Λ that have a common area | Ω | of the Wigner-Seitz cell. In (3.42), V p is thesolution to (3.7 c) and b = R ∞ w ρ dρ where w ( ρ ) > is the ground-state solution of ∆ ρ w − w + w = 0 . Numericalcomputations yield b ≈ . and R ∞ V p ρ dρ ≈ . . The numerical method to compute K s is given in §
6. In § K s is maximized for a regular hexagonal lattice. Thus, the maximal stability threshold for D isobtained for a regular hexagonal lattice arrangement of spots. In this section we analyze the linear stability of a steady-state periodic pattern of spots for the GM model (1.2),where the spots are centered at the lattice points of the Bravais lattice (2.1). he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems We first use the method of matched asymptotic expansions to construct a steady-state one-spot solution to (1.2)centered at the origin of the Wigner-Seitz cell Ω.In the inner region near the origin of Ω we look for a locally radially symmetric steady-state solution of the form(4.1) u = D U , v = DV , y = ε − x . Then, substituting (4.1) into the steady-state equations of (1.2), we obtain that V ∼ V ( ρ ) and U ∼ U ( ρ ), with ρ = | y | , satisfy the core problem∆ ρ V − V + V /U = 0 , ∆ ρ U = − V , < ρ < ∞ , (4.2 a ) U ′ (0) = V ′ (0) = 0 ; V → , U ∼ − S log ρ + χ ( S ) + o (1) , as ρ → ∞ , (4.2 b )where ∆ ρ V ≡ V ′′ + ρ − V ′ and S = R ∞ V ρ dρ . The unknown source strength S will be determined by matching thefar-field behavior of the core solution to an outer solution for u valid away from O ( ε ) distances from the origin.Since v is exponentially small in the outer region, we have in the sense of distributions that ε − v → πD Sδ ( x ).Therefore, from (1.2), the outer steady-state problem for u is∆ u − D u = − πDS δ ( x ) , x ∈ Ω ; P u = 0 , x ∈ ∂ Ω ,u ∼ − DS log | x | + D (cid:18) − Sν + χ ( S ) (cid:19) , as x → , (4.3)where ν ≡ − / log ε . We introduce the reduced-wave Green’s function G p ( x ) and its regular part R p , which satisfy∆ G p − D G p = − δ ( x ) , x ∈ Ω ; P G p = 0 , x ∈ ∂ Ω ,G p ( x ) ∼ − π log | x | + R p , as x → , (4.4)where R p is the regular part of G p . The solution to (4.3) is u ( x ) = 2 πDSG p ( x ). Now as x → u ( x ) and compare it with the required behavior in (4.3). This yields that S satisfies(4.5) (1 + 2 πνR p ) S = νχ ( S ) . Since the stability threshold occurs when D = O ( ν − ) ≫
1, we expand the solution to (4.4) for D = D /ν ≫ D = O (1) to obtain(4.6) G p = D | Ω | ν + G p + O ( ν ) , R p = D | Ω | ν + R p + O ( ν ) , where G p and R p is the periodic source-neutral Green’s function and its regular part, respectively, defined by (3.6).By combining (4.5) and (4.6), we get that S satisfies(4.7) (cid:0) µ + 2 πνR p + O ( ν ) (cid:1) S = νχ ( S ) , µ ≡ πD | Ω | . To determine the appropriate scaling for S in terms of ν ≪ χ ( S ) = O ( S / ) as S → S = O ( ν ) as ν →
0. Thenext result determines a two-term expansion for the solution to the core problem (4.2) for ν → S = O ( ν ). Lemma 4.1
For S = S ν + S ν + · · · , where ν ≡ − / log ε ≪ , the asymptotic solution to the core problem (4.2) D. Iron, J. Rumsey, M. J. Ward, J. Weiis (4.8 a ) V ∼ ν ( V + νV + · · · ) , U ∼ ν (cid:0) U + νU + ν U + · · · (cid:1) , χ ∼ ν ( χ + νχ + · · · ) , where U , U ( ρ ) , V ( ρ ) , and V ( ρ ) are defined by (4.8 b ) U = χ , U = χ + S U p , V = χ w , V = χ w + S V p . Here w ( ρ ) is the unique ground-state solution to ∆ ρ w − w + w = 0 with w (0) > , w ′ (0) = 0 , and w → as ρ → ∞ .In terms of w ( ρ ) , the functions U p and V p are the unique solutions on ≤ ρ < ∞ to L V p = w U p , V ′ p (0) = 0 , V p → , as ρ → ∞ , ∆ ρ U p = − w /b , U ′ p (0) = 0 , U p → − log ρ + o (1) , as ρ → ∞ ; b ≡ ∞ Z ρw dρ , (4.8 c ) where L V p ≡ ∆ ρ V p − V p + 2 wV p . Finally, in (4.8 a), the constants χ and χ are related to S and S by (4.8 d ) χ = r S b , χ = S χ b − S b ∞ Z wV p ρ dρ . The derivation of this result is given in Appendix B below. The o (1) condition in the far-field behavior in (4.8 c )eliminates an otherwise arbitrary constant in the determination of U p . Therefore, this condition ensures that thesolution to the linear BVP (4.8 c ) is unique. We linearize around the steady-state solution u e and v e , as calculated in § P kkk is the quasi-periodic boundary operator of (2.35): ε ∆ φ − φ + 2 v e u e φ − v e u e η = λφ , x ∈ Ω ; P kkk φ = 0 , x ∈ ∂ Ω ,D ∆ η − η + 2 ε − v e φ = λτ η , x ∈ Ω ; P kkk η = 0 , x ∈ ∂ Ω . (4.9)In the inner region near x = 0 we introduce the local variables N ( ρ ) and Φ( ρ ) by(4.10) η = N ( ρ ) , φ = Φ( ρ ) , ρ = | y | , y = ε − x . Upon substituting (4.10) into (4.9), and by using u e ∼ DU and v e ∼ DV , where U and V satisfy the core problem(4.2), we obtain on 0 < ρ < ∞ that∆ ρ Φ − Φ+ 2 VU Φ − V U N = λ Φ , Φ → , as ρ → ∞ , ∆ ρ N = − V Φ , N ∼ − C log ρ + B , as ρ → ∞ , (4.11)with Φ ′ (0) = N ′ (0) = 0 and where B = B ( S ; λ ). The divergence theorem yields the identity C = 2 R ∞ V Φ ρ dρ .To determine the constant C we must match the far field behavior of the core solution to an outer solution for η , which is valid away from x = 0. Since v e is localized near x = 0, we calculate in the sense of distributions that2 ε − v e φ → D (cid:0)R R V Φ d y (cid:1) δ ( x ) = 2 πCDδ ( x ). By using this expression in (4.9), we obtain that the outer problem he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems η is ∆ η − θ λ η = − πCδ ( x ) , x ∈ Ω ; P kkk η = 0 , x ∈ ∂ Ω ,η ∼ − C log | x | − Cν − B , as x → , (4.12)where we have defined θ λ ≡ p (1 + τ λ ) /D . The solution to (4.12) is η = 2 πC G bλ ( x ), where G bλ satisfies∆ G bλ − θ λ G bλ = − δ ( x ) , x ∈ Ω ; P kkk G bλ = 0 , x ∈ ∂ Ω , G bλ ∼ − π log | x | + R bλ , as x → . (4.13)By imposing that the behavior of η as x → πν R bλ ) C = νB . Then,since D = D /ν ≫
1, we have from Lemma 2.3(ii), upon taking the D ≫ R bλ ∼ R b + O ( ν )for | kkk | > kkk/ (2 π ) ∈ Ω B . This yields,(4.14) (cid:0) πνR b + O ( ν ) (cid:1) C = νB , where R b = R b ( kkk ) is the regular part of the Bloch Green’s function G b defined on Ω by (2.12).As in § O ( ν ) neighborhood of the origin λ = 0 when D is close to a certain critical value. To do so, we first mustcalculate an asymptotic expansion for the solution to (4.11) together with (4.14).By using (4.8 a ) we first calculate the coefficients in the differential operator in (4.11) as VU = w + νS χ ( V p − wU p ) + · · · , V U = w + 2 νS χ w ( V p − wU p ) + · · · , so that the local problem (4.11) on 0 < ρ < ∞ becomes∆ ρ Φ − Φ + (cid:20) w + 2 νS χ w ( V p − wU p ) + · · · (cid:21) Φ = (cid:20) w + 2 νS χ w ( V p − wU p ) + · · · (cid:21) N + λ Φ , ∆ ρ N = − ν [ χ w + ν ( χ w + S V p ) + · · · ] Φ , Φ → , N ∼ − C log ρ + B , as ρ → ∞ ; Φ ′ (0) = N ′ (0) = 0 . (4.15)To analyze (4.15) together with (4.14), we substitute the appropriate expansions N = 1 ν (cid:16) ˆ N + ν ˆ N + · · · (cid:17) , B = 1 ν (cid:16) ˆ B + ν ˆ B + · · · (cid:17) , C = C + νC + · · · , Φ = 1 ν (Φ + ν Φ + · · · ) , λ = λ + νλ + · · · , (4.16)into (4.15) and collect powers of ν .To leading order, we obtain on 0 < ρ < ∞ that L Φ ≡ ∆ ρ Φ − Φ + 2 w Φ = w ˆ N + λ Φ , ∆ ρ ˆ N = 0 , Φ → , ˆ N → ˆ B as ρ → ∞ ; Φ ′ (0) = ˆ N ′ (0) = 0 , (4.17)where L is the local operator. We conclude that ˆ N = ˆ B for ρ ≥ ρ > satisfies(4.18) L Φ − w ˆ N = − S χ ( V p − wU p ) Φ + 2 S χ w ( V p − wU p ) ˆ N + λ Φ ; Φ → , as ρ → ∞ , with Φ ′ (0) = 0, and that ˆ N on ρ > ρ ˆ N = − χ w Φ ; ˆ N ∼ − C log ρ + ˆ B , as ρ → ∞ ; ˆ N ′ (0) = 0 . D. Iron, J. Rumsey, M. J. Ward, J. Wei
At one higher order, the problem for ˆ N on ρ > ρ ˆ N = − χ w Φ − χ w + S V p ) Φ ; ˆ N ∼ − C log ρ + ˆ B , as ρ → ∞ ; ˆ N ′ (0) = 0 . In addition, by substituting (4.16) into (4.14) we obtain, upon collecting powers of ν , that(4.21) C = ˆ B , C + 2 πR b ˆ B = ˆ B . Next, we proceed to analyze (4.17)–(4.20). From the divergence theorem, we obtain from (4.19) that(4.22) C = 2 χ ∞ Z w Φ ρ dρ . To identify χ in (4.22), we substitute S = ν S + · · · and χ ∼ νχ + · · · into (4.7) to get (1 + µ + · · · ) (cid:0) ν S + · · · (cid:1) ∼ ν ( χ + · · · ). From the leading order terms, we get χ = S (1 + µ ). Then, since S = bχ from (4.8 d ), we obtain(4.23) χ = 1 b (1 + µ ) , S = 1 b (1 + µ ) , C = ˆ B = ˆ N = 2 b (1 + µ ) ∞ Z w Φ ρ dρ . From (4.17) we then obtain the leading-order NLEP on ρ > L Φ − w (1 + µ ) R ∞ w Φ ρ dρ R ∞ w ρ dρ = λ Φ ; Φ → , as ρ → ∞ ; Φ ′ (0) = 0 ; µ ≡ πD | Ω | . For this NLEP, Theorem 3.7 of [ ] proves that Re( λ ) < / (1 + µ ) >
1. Therefore, the stabilitythreshold where λ = 0 and Φ = w occurs when µ = 1. At this stability threshold, we calculate from (4.23) that(4.25) χ = 12 b , S = 14 b , Φ = w , C = ˆ B = ˆ N = 1 b ∞ Z w Φ ρ dρ = 1 . Upon substituting (4.25) into (4.19) we obtain at λ = 0 that ˆ N on ρ > ρ ˆ N = − χ w = − w b ; ˆ N ∼ − log ρ + ˆ B , as ρ → ∞ ; ˆ N ′ (0) = 0 . Upon comparing (4.26) with the problem for U p in (4.8 c ), we conclude that(4.27) ˆ N = U p + ˆ B . As in § O ( ν ) ball near λ = 0 when the bifurcation parameter µ is O ( ν ) close to theleading-order critical value µ = 1. As such, we set(4.28) λ = νλ + · · · , for µ = 1 + νµ + · · · , and we derive an expression for λ in terms of the de-tuning parameter µ , the Bloch wavevector kkk , the latticestructure, and certain correction terms to the core problem.We first use (4.8 d ) and (4.7) to calculate χ in terms of µ . By substituting µ = 1 + νµ + · · · together with (4.8 a )into (4.7), we obtain[1 + (1 + νµ ) + 2 πνR p + · · · ] (cid:2) ν S + ν S + · · · (cid:3) = ν ( χ + νχ + · · · ) . From the O ( ν ) terms, we obtain that χ = µ S + 2 S + 2 πR p S . Upon combining this result together with (4.8 d ) he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems χ , and by using χ = 1 / (2 b ), we obtain at criticality where λ = 0 that(4.29) χ = − µ b − πR p b − b ∞ Z wV p ρ dρ . This result is needed below in the evaluation of the solvability condition.Next, we substitute (4.25) and (4.27) into (4.18) for Φ to obtain, after some algebra, that (4.18) reduces at theleading-order stability threshold λ = 0 to(4.30) L Φ − w ˆ B = λ w + w U p ; Φ → , as ρ → ∞ , with Φ ′ (0) = 0. In a similar way, at the leading-order stability threshold λ = 0, the problem (4.20) for ˆ N on ρ > ρ ˆ N = − wb Φ − (cid:18) χ w + 14 b V p (cid:19) w ; ˆ N ∼ − C log ρ + ˆ B , as ρ → ∞ ; ˆ N ′ (0) = 0 . By applying the divergence theorem to (4.31) we get(4.32) C = 1 b ∞ Z w Φ ρ dρ + 2 χ b + 12 b ∞ Z wV p ρ dρ . Then, by using (4.21) with ˆ B = 1 to relate C to ˆ B , we determine ˆ B as ˆ B = C + 2 πR b where C is given in(4.32). With ˆ B obtained in this way, we find from (4.30) that Φ satisfies(4.33 a ) L Φ ≡ L Φ − w R ∞ w Φ ρ dρ R ∞ w ρ dρ = R g + λ w ; Φ → , as ρ → ∞ , with Φ ′ (0) = 0, where the residual R g is defined by(4.33 b ) R g ≡ πR b w + 2 χ bw + 12 b w ∞ Z wV p ρ dρ + w U p . As discussed in § a ) is orthogonal to thehomogeneous adjoint solution Ψ ⋆ = w + ρw ′ / λ R ∞ w Ψ ⋆ ρ dρ + R ∞ R g Ψ ∗ ρ dρ = 0. Upon using(4.29), which relates χ to µ , to simplify this solvability condition, we readily obtain by using (4.33 b ) for R g that(4.34) λ = − R ∞ w Ψ ⋆ ρ dρ R ∞ w Ψ ⋆ ρ dρ πR b − µ − πR p − b ∞ Z wV p ρ dρ − R ∞ w U p Ψ ⋆ ρ dρ R ∞ w Ψ ⋆ ρ dρ . To simplify the terms in (4.34), we use L V p = w U p and ∆ ρ U p = − w /b from (4.8 c ), together with w = L − Ψ ⋆ to calculate, after an integration by parts, that ∞ Z w U p Ψ ⋆ ρ dρ = ∞ Z ( L V p ) (cid:0) L − w (cid:1) ρ dρ = ∞ Z V p wρ dρ . By substituting this expression, together with R ∞ w Ψ ⋆ ρ dρ = b and R ∞ w Ψ ⋆ ρ dρ = b/
2, as obtained from (3.38),into (4.34) we obtain our final result for λ . We summarize our result as follows: Principal Result 4.1
In the limit ε → , consider a steady-state periodic pattern of spots for the GM model (1.2) D. Iron, J. Rumsey, M. J. Ward, J. Weiwhere D = O ( ν − ) with ν = − / log ε . Then, when (4.35 a ) D ∼ | Ω | πν (1 + νµ ) , where µ = O (1) and | Ω | is the area of the Wigner-Seitz cell, the portion of the continuous spectrum of the linearizationthat lies within an O ( ν ) neighborhood of the origin λ = 0 is given by (4.35 b ) λ = νλ + · · · , λ = 2 µ − πR b + πR p − b ∞ Z ρwV p dρ . Here R b = R b ( kkk ) is the regular part of the Bloch Green’s function G b defined on Ω by (2.12), kkk/ (2 π ) ∈ Ω B , and R p is the regular part of the periodic source-neutral Green’s function G p satisfying (3.6). Remark 4.1
In comparison with the analogous result obtained in Principal Result 3.1 for the Schnakenburg model, λ in (4.35 b) now depends on the regular parts of two different Green’s functions. The term R p only depends on thegeometry of the lattice, whereas R b = R b ( kkk ) depends on both the lattice geometry and the Bloch wavevector kkk . Tocalculate R b ( kkk ) we again need only consider vectors kkk/ (2 π ) in the first Brillouin zone Ω B of the reciprocal lattice.Since R b is real-valued from Lemma 2.1, the band of spectrum (4.35 b) lies on the real axis in the λ -plane. Moreover,from Lemma 2.2 small values of | kkk | generate spectra that lie at an O (cid:16) ν/kkk T Q kkk (cid:17) distance from the origin along thenegative real axis in the λ -plane, where Q is a positive definite matrix. For a given lattice geometry, we seek to determine µ so that λ < kkk . From (4.35 b ), we conclude that aperiodic arrangement of spots with a given lattice structure is linearly stable when(4.36) µ < πR ⋆b − πR p + 1 b ∞ Z wV p ρ dρ , R ⋆b ≡ min kkk R b ( kkk ) . We characterize the optimal lattice as the one with a fixed area | Ω | of the Wigner-Seitz cell that allows for stabilityfor the largest inhibitor diffusivity D . This leads to our second main result. Principal Result 4.2
The optimal lattice arrangement for a periodic pattern of spots for the GM model (1.2) isthe one for which the objective function K gm is maximized, where (4.37) K gm ≡ πR ⋆b − πR p , R ⋆b ≡ min kkk R b ( kkk ) . For ν = − / log ε ≪ , a two-term asymptotic expansion for this maximal stability threshold for D is given explicitlyby (4.38) D optim ∼ | Ω | πν ν max Λ K gm + 1 b ∞ Z wV p ρ dρ , where max Λ K gm is taken over all lattices Ω having a common area | Ω | of the Wigner-Seitz cell. In (4.38), V p is thesolution to (4.8 c) and b = R ∞ w ρ dρ ≈ . where w ( ρ ) > is the ground-state solution of ∆ ρ w − w + w = 0 .Numerical computations yield R ∞ wV p ρ dρ ≈ − . . The numerical method to compute K gm is given in §
6. In § D occurs for a regular hexagonal lattice. he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems In this section we implement a very simple alternative approach to calculate the stability threshold for the Schnaken-burg (1.1) and GM Models (1.2) in § § § µ is O ( ν ) close to its critical value.Instead, we determine the critical value of µ , depending on the Bloch wavevector kkk , such that λ = 0 is in the spectrumof the linearization. We then perform a min-max optimization of this critical value of µ with respect to kkk and thelattice geometry Λ in order to find the optimal value of D . This alternative approach for calculating the stability threshold requires the following two-term expansion for χ ( S )in terms of S as S → Lemma 5.1
For S → , the asymptotic solution to the core problem (3.2) is V ∼ Sb w + S b ( − ˆ χ bw + V p ) + · · · , U ∼ bS + S (cid:18) ˆ χ + U p b (cid:19) + · · · ,χ ∼ bS + S ˆ χ + · · · ; ˆ χ ≡ b ∞ Z V p ρ dρ . (5.1) Here w ( ρ ) is the unique positive ground-state solution to ∆ ρ w − w + w = 0 and b ≡ R ∞ w ρdρ . In terms of w ( ρ ) ,the functions U p and V p are the unique solutions on ≤ ρ < ∞ to (3.7 c). The derivation of this result, as outlined at the end of Appendix A, is readily obtained by setting S = 0 and S = S ν / in the results of Lemma 3.1.The key step in the analysis is to note that at λ = 0, the solution to the inner problem (3.11) for Φ and N can bereadily identified by differentiating the core problem (3.2) with respect to S . More specifically, at λ = 0, the solutionto (3.11) is Φ = CV S , N = CU S , and B ( S ; 0) = Cχ ′ ( S ). With B known at λ = 0, we obtain from (3.15) and (3.4)that the critical value of D at λ = 0 satisfies the nonlinear algebraic problem(5.2) 1 + 2 πνR b + O ( ν ) + νχ ′ ( S ) = 0 , where S = a | Ω | π √ D .
To determine the critical threshold in D from (5.2) we use the two-term expansion for χ ( S ) in (5.1) to get χ ′ ( S ) ∼ − b/S + ˆ χ + · · · . By using the relation for S in terms of D from (5.2) when D = D /ν ≫
1, we obtain that(5.3) χ ′ ( S ) ∼ − µν + ˆ χ + · · · , µ ≡ π D ba | Ω | , D = D ν . Upon substituting this expression into (5.2), we obtain that1 − µ + ν ˆ χ = − πνR b + O ( ν ) , which determines µ as µ ∼ ν (2 πR b + ˆ χ ). Upon recalling the definition of µ in (5.3), we conclude that λ = 0when D = D ⋆ ( kkk ), where D ⋆ ( kkk ) is given by(5.4) D ⋆ ( kkk ) ≡ a | Ω | π bν (cid:2) ν (2 πR b ( kkk ) + ˆ χ ) + O ( ν ) (cid:3) , D. Iron, J. Rumsey, M. J. Ward, J. Wei where ˆ χ is defined in (5.1). By minimizing R b ( kkk ) with respect to kkk , and then maximizing the result with respect tothe geometry of the lattice Λ, (5.4) recovers the main result (3.42) of Principal Result 3.2. This simple method, whichrelies critically on the observation that B = χ ′ ( S ) at λ = 0, provides a rather expedient approach for calculating theoptimal threshold in D . However, it does not characterize the spectrum contained in the small ball | λ | = O ( ν ) ≪ D is near the leading-order stability threshold a | Ω | / (4 π bν ). Next, we use a similar approach as in § S for χ ( S ) as S → Lemma 5.2
For S → , the asymptotic solution to the core problem (4.2) is V ∼ r Sb w + S ( ˆ χ w + V p ) + · · · , U ∼ r Sb + S ( ˆ χ + U p ) + · · · ,χ ∼ r Sb + S ˆ χ + · · · , ˆ χ ≡ − b ∞ Z wV p ρ dρ . (5.5) Here w ( ρ ) is the unique positive ground-state solution to ∆ ρ w − w + w = 0 and b ≡ R ∞ w ρdρ . In terms of w ( ρ ) ,the functions U p and V p are the unique solutions on ≤ ρ < ∞ to (4.8 c). The derivation of this result, as outlined at the end of Appendix B, is readily obtained by setting S = 0 and S = S ν in the results of Lemma 4.1.As similar to the analysis in § N is readily identified by differentiating the coreproblem (4.2) with respect to S . In this way, we get B ( S,
0) = Cχ ′ ( S ). Therefore, at λ = 0, we obtain from (4.14)and (4.7) that the critical values of D and S where λ = 0 satisfy the coupled nonlinear algebraic system (cid:0) µ + 2 πνR p + O ( ν ) (cid:1) S = νχ ( S ) , µ ≡ πD | Ω | , D = D ν , πνR b + O ( ν ) − νχ ′ ( S ) = 0 . (5.6)We then use the two term expansion in (5.5) for χ ( S ) as S → § S must first be calculated from a nonlinear algebraic equation. By substituting (5.5) for χ ( S ) into the firstequation of (5.6), and expanding µ = µ + νµ + · · · , we obtain[1 + µ + ν ( µ + 2 πνR p )] S ∼ ν r Sb + S ˆ χ ! , which can be solved asymptotically when ν ≪ S in terms of µ and µ given by(5.7) S = ν (cid:16) ˆ S + ν ˆ S + · · · (cid:17) ; ˆ S ≡ b (1 + µ ) , ˆ S ≡ b (1 + µ ) ( ˆ χ − µ − πR p ) . From the two-term expansion (5.7) for S we calculate χ ′ ( S ) from (5.5) as χ ′ ( S ) ∼ √ bν (cid:16) ˆ S + ν ˆ S + · · · (cid:17) − / + ˆ χ ∼ ˆ S − / √ bν + " ˆ χ − ˆ S √ b ˆ S / + O ( ν ) . he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems S and ˆ S , the expression above becomes(5.8) χ ′ ( S ) ∼ ν (cid:2) (1 + µ ) + ν ( ˆ χ + µ + 2 πR p ) + O ( ν ) (cid:3) . Then, upon substituting (5.8) into the second equation of (5.6) we obtain, up to O ( ν ) terms, that1 + 2 πνR b ∼ (1 + µ ) + ν χ + 2 πR p + µ ) , which determines µ and µ as(5.9) µ = 1 , µ = − ˆ χ − πR p + 4 πR b . Finally, by recalling the definition of µ and ˆ χ in (5.6) and (5.5), respectively, and by using the two-term expansion µ = µ + νµ from (5.9), we conclude that λ = 0 when D = D ⋆ ( kkk ), where D ⋆ ( kkk ) is given by(5.10) D ⋆ ( kkk ) ≡ | Ω | πν ν πR b ( kkk ) − πR p + 1 b ∞ Z wV p ρ dρ + O ( ν ) . By minimizing R b ( kkk ) with respect to kkk , and then maximizing the result with respect to the geometry of the latticeΛ, (5.10) recovers the main result (4.38) of Principal Result 4.2. In this sub-section we employ the simple approach of § § ] is(5.11) v t = ε ∆ v − v + Auv , τ u t = D ∆ u + (1 − u ) − uv , x ∈ Ω ; P u = P v = 0 , x ∈ ∂ Ω , where ε > D > τ >
1, and the feed-rate parameter
A > A and D , the stability and self-replication behavior of localized spots for (5.11) have been studied in [ ], [ ], [ ], [ ],and [ ] (see also the references therein). We will consider the parameter regime D = O ( ν − ) ≫ A = O ( ε )of [ ]. In this regime, and to leading order in ν , an existence and stability analysis of N -spot patterns in a finitedomain was undertaken via a Lypanunov Schmidt reduction and a rigorous study of certain nonlocal eigenvalueproblems. We briefly review the main stability result of [ ] following (5.16 b ) below.We first construct a one-spot steady-state solution to (5.11) with spot centered at x = 0 in Ω in the regime D = O ( ν − ) and A = O ( ε ) by using the approach in § ].In the inner region near x = 0 we introduce the local variables U , V , and y , defined by(5.12) u = εA √ D U , v = √ Dε V , y = ε − x , into the steady-state problem for (5.11). We obtain that U ∼ U ( ρ ) and V ∼ V ( ρ ), with ρ = | y | , satisfy the samecore problem ∆ ρ V − V + U V = 0 , ∆ ρ U − U V = 0 , < ρ < ∞ , (5.13 a ) U ′ (0) = V ′ (0) = 0 ; V → , U ∼ S log ρ + χ ( S ) + o (1) , as ρ → ∞ , (5.13 b )as that for the Schnakenburg model studied in § S ≡ R ∞ U V ρ dρ and ∆ ρ V ≡ V ′′ + ρ − V ′ . Therefore, for S →
0, the two-term asymptotics of χ ( S ) is given in (5.1) of Lemma 5.1.8 D. Iron, J. Rumsey, M. J. Ward, J. Wei
To formulate the outer problem for u , we observe that since v is localized near x = 0 we have in the sense ofdistributions that uv → ε (cid:16)R R √ D ( Aε ) − U V d y (cid:17) δ ( x ) ∼ πε √ DA − S δ ( x ). Then, upon matching u to the coresolution U , we obtain from (5.11) that∆ u + 1 D (1 − u ) = 2 π εA √ D S δ ( x ) , x ∈ Ω ; P u = 0 , x ∈ ∂ Ω ,u ∼ εA √ D (cid:18) S log | x | + Sν + χ ( S ) (cid:19) , as x → , (5.14)where ν ≡ − / log ε . The solution to (5.14) is u = 1 − πεSG p ( x ) / ( A √ D ), where G p ( x ) is the Green’s function of(4.4). Next, we calculate the local behavior of u as x → S satisfies(5.15) S + ν [ χ ( S ) + 2 πSR p ] = Aν √ Dε , where R p is the regular part of G p as defined in (4.4).We consider the regime D = D /ν ≫ D = O (1). By using the two-term expansion (4.6) for R p in termsof the regular part R p of the periodic source-neutral Green’s of (3.6), (5.15) becomes(5.16 a ) S (1 + µ ) + ν [2 πSR p + χ ( S )] + O ( ν ) = A√ νµ , where we have defined µ and A = O (1) in terms of A = O ( ε ) by(5.16 b ) A = Aε r | Ω | π , µ ≡ πD | Ω | , D = D ν . To illustrate the bifurcation diagram associated with (5.16 a ), we use χ ( S ) ∼ b/S as S → S = ν / S , with S = O (1), we obtain from (5.16) that, to leading order in ν ,(5.17) A√ µ = S (1 + µ ) + b S ; µ = 2 πD | Ω | , b = ∞ Z w ρ dρ . From Lemma 5.1 and (5.12), the spot amplitude V (0) to leading order in ν is related to S by V (0) = ε − √ D S w (0) /b .In Fig. 4 we use (5.17) to plot the leading-order saddle-node bifurcation diagram of S versus A , where the upper solu-tion branch corresponds a pattern with large amplitude spots. The saddle-node point occurs when S f = p b/ (1 + µ )and A f = 2 √ b p (1 + µ ) /µ . As we show below, there is a zero eigenvalue crossing corresponding to an instabilityfor some Bloch wavevector kkk with | kkk | > kkk/ (2 π ) ∈ Ω B that occurs within an O ( ν ) neighborhood of the point( S , A ) on the upper branch of Fig. 4 given by S = √ b and A = (2 + µ ) p b/µ . Since | kkk | > A = A + ν A + · · · , and determine the optimallattice arrangement of spots that minimizes A . This has the effect of maximizing the extent of the upper solutionbranch in Fig. 4 that is stable to competition instabilities.Before proceeding with the calculation of the optimal lattice for the periodic problem, we recall some prior rigorousresults of [ ] for the finite domain problem with N localized spots in a finite domain Ω N with homogeneous Neumannboundary conditions. From [ ], the bifurcation diagram to leading order in ν is A√ µ N = S (1 + µ N ) + b S , b = ∞ Z w ρ dρ , µ N = 2 πN D | Ω N | , A = Aε r | Ω N | πN , which shows that we need only replace | Ω | in (5.17) with | Ω N | /N . From a rigorous NLEP analysis of the finite domain he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems . . . . . . S A
Figure 4. Plot, to leading-order in ν , of the saddle-node bifurcation diagram S versus A , obtained from (5.17), forthe GS model with | Ω | = 1 and D = 1. The leading-order spot amplitude V (0) = ε − √ D S w (0) /b is directlyproportional to S . The heavy solid branch of large amplitude spots is linearly stable to competition instabilities,while the dotted branch is unstable to competition instabilities. To leading order in ν , the zero eigenvalue crossingcorresponding to the competition instability threshold occurs at A = (2 + µ ) p b/µ ≈ .
34 where S = √ b ≈ . ] that the lower solution branch in Fig. 4 is unstable to synchronous instabilities, whilethe upper branch is stable to such instabilities. In contrast, it is only the portion of the upper solution branch with S > S that is stable to competition instabilities (see Fig. 4). Therefore, there is two zero eigenvalue crossings; one atthe saddle-node point corresponding to a synchronous instability, and one at the point ( S , A ) on the upper branchcorresponding to a competition instability.Similarly, for the periodic spot problem we remark that there is also a zero eigenvalue crossing when kkk = 0, i.e. thesynchronous instability, which occurs at the saddle-node bifurcation point. However, since it is the zero eigenvaluecrossing for the competition instability that sets the instability threshold for A (see Fig. 4), we will not analyze theeffect of the lattice geometry on the zero eigenvalue crossing for the synchronous instability mode.We now proceed to analyze the zero eigenvalue crossing for the competition instability. To determine the stabilityof the steady-state solution u e and v e , we introduce (3.8) into (5.11) to obtain the Floquet-Bloch eigenvalue problem ε ∆ φ − φ + 2 Au e v e φ + Av e η = λφ , x ∈ Ω ; P kkk φ = 0 , x ∈ ∂ Ω ,D ∆ η − η − u e v e φ − v e η = λτ η ; P kkk φ = 0 , x ∈ ∂ Ω . (5.18)In the inner region near x = 0 we look for a locally radially symmetric eigenpair N ( ρ ) and Φ( ρ ), with ρ = | y | ,defined in terms of η and φ by(5.19) η = εA √ D N ( ρ ) , φ = √ Dε Φ( ρ ) , ρ = | y | , y = ε − x . From (5.18), we obtain to within negligible O ( ε ) terms that N ( ρ ) and Φ( ρ ) satisfy∆ ρ Φ − Φ+2
U V
Φ +
N V = λ Φ , Φ → , as ρ → ∞ , ∆ ρ N = 2 U V
Φ +
N V , N ∼ C log ρ + B , as ρ → ∞ , (5.20)with Φ ′ (0) = N ′ (0) = 0, B = B ( S ; λ ), and C = R ∞ (cid:0) U V
Φ +
N V (cid:1) ρ dρ .To determine the outer problem for η , we first calculate in the sense of distributions that(5.21) 2 u e v e φ + v e η → √ DAε ε Z R (cid:0) U V
Φ + V N (cid:1) d y δ ( x ) = 2 πε √ DA Cδ ( x ) . D. Iron, J. Rumsey, M. J. Ward, J. Wei
Then, by asymptotically matching η as x → N in (5.20), we obtain from (5.21) and(5.18) that the outer problem for η is∆ η − θ λ η = 2 πεA √ D Cδ ( x ) , x ∈ Ω ; P kkk η = 0 , x ∈ ∂ Ω ,η ∼ εA √ D (cid:20) C log | x | + Cν B (cid:21) , as x → . (5.22)Here we have defined θ λ ≡ p (1 + τ λ ) /D . The solution to (5.22) is η = − πεC G bλ ( x ) / ( A √ D ), where G bλ satisfies(4.13). By imposing that the behavior of η as x → πν R bλ ) C + νB = 0, where R bλ is the regular part of G bλ defined in (4.13). Then, since D = D /ν ≫
1, we have from Lemma2.3(ii) upon taking the D ≫ R bλ ∼ R b + O ( ν ) for | kkk | > kkk/ (2 π ) ∈ Ω B . Thus, we have(5.23) (cid:0) πνR b + O ( ν ) (cid:1) C − νB = 0 , where R b = R b ( kkk ) is the regular part of the Bloch Green’s function G b defined on Ω by (2.12). We remark thatif we were to consider zero-eigenvalue crossings for a synchronous instability where kkk = 0, we would instead use R bλ = R p ∼ D /ν | Ω | + R p + · · · from (4.6) to obtain (cid:0) µ + 2 πνR p + O ( ν ) (cid:1) C + νB = 0 in place of (5.23).As in § λ = 0, we have B ( S ; 0) = Cχ ′ ( S ). Therefore, at λ = 0, we obtain from (5.23)and (5.16 a ) that the critical values of A and S where λ = 0 satisfy the coupled nonlinear algebraic system(5.24) S (1 + µ ) + ν [2 πSR p + χ ( S )] + O ( ν ) = A√ νµ , πνR b + O ( ν ) + νχ ′ ( S ) = 0 . The final step in the calculation is to use the two term expansion for χ ( S ), as given in (5.1) of Lemma 5.1, toobtain a two-term approximate solution in powers of ν to (5.24). By substituting χ ′ ( S ) ∼ − bS − + ˆ χ for S ≪ S as(5.25) S ∼ √ bν (cid:16) ν ˆ S + · · · (cid:17) , ˆ S ≡ −
12 ( ˆ χ + 2 πR b ) . Then, we substitute (5.25), together with the two-term expansion(5.26) A = A + ν A + · · · , into the first equation of (5.24), and equate powers of ν . From the O ( ν / ) terms in the resulting expression weobtain that A = √ b (2 + µ ) / √ µ , while at order O ( ν / ) we get that A = A ( kkk ) satisfies(5.27) A √ bµ = 2 πR p µ + ˆ χ µ + ˆ S = 2 πR p µ − πR b ( kkk ) + ˆ χ (2 − µ )2 µ , where ˆ χ is given in (5.1) of Lemma 5.1.To determine the optimal lattice that allows for stability for the smallest value of A , we first fix a lattice Λ andthen maximize A in (5.27) through minimizing R b ( kkk ) with respect to the Bloch wavevector kkk . Then, we minimize A with respect to the lattice geometry Λ while fixing | Ω | . We summarize this third main result as follows: Principal Result 5.1
The optimal lattice arrangement for a steady-state periodic pattern of spots for the GS model(5.11) in the regime D = D /ν ≫ and A = O ( ε ) is the one for which the objective function K gs is maximized,where (5.28) K gs ≡ πµR ⋆b − πR p , R ⋆b ≡ min kkk R b ( kkk ) , µ ≡ πD | Ω | . For ν = − / log ε ≪ , a two-term asymptotic expansion for the competition instability threshold of A on the optimalhe Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems lattice is (5.29) A optim = ε s π | Ω | A optim , A optim ∼ √ b (2 + µ ) √ µ + ν s bµ − max Λ K gs + 1 b (cid:16) − µ (cid:17) ∞ Z V p ρ dρ + · · · , where max Λ K gs is taken over all lattices Λ having a common area | Ω | of the Wigner-Seitz cell. In (5.29), V p is thesolution to (3.7 c), while b = R ∞ w ρ dρ ≈ . , where w ( ρ ) > is the ground-state solution of ∆ ρ w − w + w = 0 ,and R ∞ V p ρ dρ ≈ . . We remark that (5.29) can also be derived through the more lengthy but systematic approach given in § § | λ | ≤ O ( ν ) when A = A + O ( ν ).The numerical method to compute K gs is given in §
6. In § K gs is maximized for a regular hexagonal lattice. Thus, the minimal stability threshold for thefeed-rate A occurs for this hexagonal lattice. We seek a rapidly converging expansion for the Bloch Green’s function G b satisfying (2.12) on the Wigner-Seitz cellΩ for the Bravais lattice Λ of (2.1). It is the regular part R b of this Green’s function that is needed in PrincipalResults 3.2, 4.2, and 5.1. Since only one Green’s function needs to be calculated numerically in this section, for clarityof notation we remove its subscript. In § § G ≡ G b on all of R that satisfies(6.1) ∆ G ( x ) = − δ ( x ) ; G ( x + lll ) = e − ikkk · lll G ( x ) , lll ∈ Λ , where kkk/ (2 π ) ∈ Ω B . The regular part R (0) ≡ R b (0) of this Bloch Green’s function is defined(6.2) R (0) ≡ lim x → (cid:18) G ( x ) + 12 π log | x | (cid:19) . To derive a computationally tractable expression for R (0) we will follow closely the methodology of [ ].We construct the solution to (6.1) as the sum of free-space Green’s functions(6.3) G ( x ) = X lll ∈ Λ G free ( x + lll ) e ikkk · lll . This sum guarantees that the quasi-periodicity condition in (6.1) is satisfied. That is, if G ( x ) = P lll ∈ Λ G free ( x + lll ) e ikkk · lll , then, upon choosing any lll ⋆ ∈ Λ, it follows that G ( x + lll ⋆ ) = e − ikkk · lll ⋆ G ( x ) . To show this, we use lllllllll ⋆ + lllllllll ∈ Λ and calculate G ( x + lll ∗ ) = X lll ∈ Λ G free ( x + lll ∗ + lll ) e ikkk · lll = X lll ∈ Λ G free ( x + lll ∗ + lll ) e ikkk · ( lll ∗ + lll ) e − ikkk · lll ∗ = e − ikkk · lll ∗ G ( x ) . In order to analyze (6.3), we will use the Poisson summation formula with converts a sum over Λ to a sum overthe reciprocal lattice Λ ⋆ of (2.5). In the notation of [ ], we have (see Proposition 2.1 of [ ])(6.4) X lll ∈ Λ f ( x + lll ) e ikkk · lll = 1 V X ddd ∈ Λ ∗ ˆ f (2 πddd − kkk ) e i x · (2 πddd − kkk ) , x , kkk ∈ R , where ˆ f is the Fourier transform of f , and V = | Ω | is the area of the primitive cell of the lattice.2 D. Iron, J. Rumsey, M. J. Ward, J. Wei
Remark 6.1
Other authors (cf. [ ], [ ]) define the reciprocal lattice as Λ ⋆ = { πm ddd , πn ddd } m,n ∈ Z , so that forany lll ∈ Λ and ddd ∈ Λ ∗ , it follows that lll · ddd = 2 Kπ for some integer K and hence e illl · ddd = 1 . The form of the Poissonsummation formula will then differ slightly from (6.4) . By applying (6.4) to (6.3), it follows that the sum over the reciprocal lattice consists of free-space Green’s functionsin the Fourier domain, and we will split each Green’s function in the Fourier domain into two parts in order to obtaina rapidly converging series. In R , we write the Fourier transform pair as(6.5) ˆ f ( ppp ) = Z R f ( x ) e − i x · ppp d x , f ( x ) = 14 π Z R ˆ f ( ppp ) e ippp · x d ppp . The free space Green’s function satisfies ∆ G free = − δ ( x ). By taking Fourier transforms, we get −| ppp | ˆ G free ( ppp ) = − G free ( ppp ) = 1 | ppp | . With the right-hand side of the Poisson summation formula (6.4) in mind, we write(6.7) 1 V X ddd ∈ Λ ∗ ˆ G free (2 πddd − kkk ) e i x · (2 πddd − kkk ) = X ddd ∈ Λ ∗ e i x · (2 πddd − kkk ) | πddd − kkk | , since V = 1. To obtain a rapidly converging series expansion, we introduce the decomposition(6.8) ˆ G free (2 πddd − kkk ) = α (2 πddd − kkk, η ) ˆ G free (2 πddd − kkk ) + (cid:16) − α (2 πddd − kkk, η ) (cid:17) ˆ G free (2 πddd − kkk ) , for some function α (2 πddd − kkk, η ). We choose α (2 πddd − kkk, η ) so that the sum over ddd ∈ Λ ∗ of the first set of termsconverges absolutely. We apply (6.5) to the second set of terms after first writing (1 − α ) ˆ G free as an integral. In thedecomposition (6.8) we choose the function α as(6.9) α (2 πddd − kkk, η ) = exp (cid:18) − | πddd − kkk | η (cid:19) , where η > η → α (2 πddd − kkk, η ) = 0 ; lim η →∞ α (2 πddd − kkk, η ) = 1 ; ∂α∂η = | πddd − kkk | α η > , since α > , η > , which shows that 0 < α < < η < ∞ . Since 0 < α <
1, the choice of η determines the portion of the Green’sfunction that is determined from the sum of terms in the reciprocal lattice Λ ∗ and the portion that is determinedfrom the sum of terms in the lattice Λ.With the expressions (6.9) for α and (6.6) for ˆ G free , we get(6.10) α (2 πddd − kkk, η ) ˆ G free (2 πddd − kkk ) e i x · (2 πddd − kkk ) = exp (cid:18) − | πddd − kkk | η (cid:19) e i x · (2 πddd − kkk ) | πddd − kkk | . Since 2 πddd − kkk = 0, which follows since kkk/ (2 π ) ∈ Ω B , the sum of these terms over ddd ∈ Λ ∗ converges absolutely.Following [ ], we define(6.11) G fourier ( x ) ≡ X ddd ∈ Λ ∗ exp (cid:18) − | πddd − kkk | η (cid:19) e i x · (2 πddd − kkk ) | πddd − kkk | . For the (1 − α ) ˆ G free term, we define ρ by ρ ≡ | πddd − kkk | , so that from (6.9), (6.6), and ˆ G free = ˆ G free ( | ppp | ), we get(6.12) (1 − α (2 πddd − kkk, η )) ˆ G free (2 πddd − kkk ) = 1 ρ (cid:16) − e − ρ / (4 η ) (cid:17) . he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems R e − ρ e s +2 s ds = − e − ρ e s / (2 ρ ), the right hand side of (6.12) can be calculated as2 − log(2 η ) Z −∞ e − ρ e s +2 s ds = 1 ρ (cid:16) − e − ρ / (4 η ) (cid:17) , so that(6.13) (1 − α (2 πddd − kkk, η )) G free (2 πddd − kkk ) = 2 ∞ Z log(2 η ) e − ρ e − s − s ds . To take the inverse Fourier transform of (6.13), we recall that the inverse Fourier transform of a radially symmetricfunction is the inverse Hankel transform of order zero (cf. [ ]), so that f ( r ) = (2 π ) − R ∞ ˆ f ( ρ ) J ( ρr ) ρ dρ . Uponusing the well-known inverse Hankel transform (cf. [ ]) ∞ Z e − ρ e − s ρ J ( ρr ) d ρ = 12 e s − r e s / , we calculate the inverse Fourier transform of (6.13) as12 π ∞ Z ∞ Z log(2 η ) e − ρ e − s − s ρ J ( ρr ) d s dρ = 1 π ∞ Z log(2 η ) e − s ∞ Z e − ρ e − s ρ J ( ρr ) d ρ ds = 12 π ∞ Z log(2 η ) e − s e s − r e s ds = 12 π ∞ Z log(2 η )) e − r e s ds . In the notation of [ ], we then define F sing ( x ) as(6.14) F sing ( x ) ≡ π ∞ Z log(2 η ) e − | x | e s ds , so that by the Poisson summation formula (6.4), we have(6.15) G spatial ( x ) ≡ X lll ∈ Λ e ikkk · lll F sing ( x + lll ) . In this way, for kkk/ (2 π ) ∈ Ω B , we write the Bloch Green’s function in the spatial domain as the sum of (6.11) and(6.15)(6.16) G ( x ) = X ddd ∈ Λ ∗ exp (cid:18) − | πddd − kkk | η (cid:19) e i x · (2 πddd − kkk ) | πddd − kkk | + 12 π X lll ∈ Λ e ikkk · lll ∞ Z log(2 η ) e − | x + lll | e s ds . From (6.11) and (6.15), it readily follows that G Fourier → η →
0, while G spatial → η → ∞ .Now consider the behaviour of the Bloch Green’s function as x →
0. From (6.11), we have(6.17) G Fourier (0) = X ddd ∈ Λ ∗ exp (cid:18) − | πddd − kkk | η (cid:19) | πddd − kkk | , for kkk/ (2 π ) ∈ Ω B , which is finite since | πddd − kkk | 6 = 0 and η < ∞ . It is also real-valued. Next, we decompose G spatial in (6.15) as(6.18) G spatial ( x ) = F sing ( x ) + X lll ∈ Λ lll = e ikkk · lll F sing ( x + lll ) . D. Iron, J. Rumsey, M. J. Ward, J. Wei x G ( x ) R ( x )(.1,.1) 1.1027-.12568 i .79138-.12568 i(.01,01) 1.4730-.012593 i .79526-.012593 i(10 − , − ) 1.8396-.0012593 i .79531-.0012593 i(10 − , − ) 2.2060-.00012593 i .79530-.00012593 i(10 − , − ) 2.5725-.000012593 i .79529-.000012593 i(10 − , − ) 2.9389-.0000012593 i .79531-.0000012593 i(10 − , − ) 3.3054-.00000012593 i .79530-.00000012593 i(10 − , − ) 3.6719-.000000012593 i .79531-.000000012593 i(10 − , − ) 4.0383-.0000000012593 i .79529-.0000000012593 i(10 − , − ) 4.4048-.00000000012593 i .79530-.00000000012593 i(10 − , − ) 4.7713-.000000000012594 i .79529-.000000000012594 i Table 1. The regular part R ( x ) of the Bloch Green’s function, as defined in (6.22), for x tending to the origin. Noticethat the imaginary part of R ( x ) becomes increasingly small as x →
0, as expected from Lemma 2.1 of § R (0) is real-valued.For the second term in (6.18), we can take the limit x → (cid:12)(cid:12)(cid:12) X lll ∈ Λ lll =0 e ikkk · lll F sing ( lll ) (cid:12)(cid:12)(cid:12) < ∞ . In contrast, F sing ( x ) is singular at x = 0. To calculate its singular behavior as x →
0, we write F sing ( x ) = F sing ( r ),with r = | x | , and convert F sing ( r ) to an exponential integral by introducing u by u = r e s / F sing ( r ) = 12 π ∞ Z log(2 η ) e − r e s ds = 14 π ∞ Z r η e − u u du = 14 π E ( r η ) , where E ( z ) = R ∞ z t − e − t dt is the exponential integral (cf. § ]). Upon using the series expansion of E ( z )(6.20) E ( z ) = − γ − log( z ) − ∞ X n =1 ( − n z n n n ! , for | arg z | < π , as given in § ], where γ = 0 . · · · is Euler’s constant, we have from (6.19) and (6.20) that(6.21) F sing ( r ) ∼ − γ π − log η π − log r π + o (1) , as r → . This shows that the Bloch Green’s function in (6.16) has the expected logarithmic singularity as x → G ( x ) = − π log | x | + R ( x ) , R ( x ) = G Fourier ( x ) + G Spatial ( x ) + 12 π log | x | . By letting x →
0, we have from (6.18), (6.21), (6.17), and (6.22), that for kkk/ (2 π ) ∈ Ω B (6.23) R (0) = X ddd ∈ Λ ∗ exp (cid:18) − | πddd − kkk | η (cid:19) | πddd − kkk | + X lll ∈ Λ lll = e ikkk · lll F sing ( lll ) − γ π − log η π , where F sing ( lll ) = E ( | lll | η ) / (4 π ).For a square lattice, with unit area of the primitive cell and with η = 2 and kkk = (sin π , cos π ), in Table 1 we givenumerical results for R ( x ) for various values of x as x →
0. The computations show that Im ( R ( x )) → x → § he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems Lattice R ⋆b R p K s K gm Square − . − . − . . − . − . − . . Table 2. Numerical values for R ⋆b = min kkk R (0), where R (0) is computed from (6.23), for the square and hexagonallattice for which | Ω | = 1. The third column is the regular part R p of the periodic source-neutral Green’s function(3.6). The last two columns are K s and K gm , as defined in Principal Results 3.2 and 4.2, respectively. Of the twolattices, the hexagonal lattice gives the largest values for K s and K gm . In this sub-section we determine the lattice that optimizes the stability thresholds given in Principal Results 3.2, 4.2,and 5.1, for the Schnakenburg, GM, and GS models, respectively. Recall that in the notation of § R b ( kkk ) = R (0),where R (0) is given in (6.23). The minimum of R (0) with respect to kkk is denoted by R ⋆b .In our numerical computations of R (0) from (6.23) we truncate the direct and reciprocal lattices Λ and Λ ∗ by thesubsets ¯Λ and ¯Λ ∗ of Λ and Λ ∗ , respectively, defined by¯Λ = (cid:8) n lll + n lll (cid:12)(cid:12) − M < n , n < M (cid:9) , ¯Λ ∗ = (cid:8) n ddd + n ddd (cid:12)(cid:12) − M < n , n < M (cid:9) , n , n ∈ Z . For each lattice, we must pick M , M and η so that G can be calculated accurately with relatively few terms inthe sum. These parameters are found by numerical experimentation. For the two regular lattices (square, hexagonal)we used ( M , M , η ) = (2 , , θ between lll and lll we took M = 5, M = 3, and we set η = 3.In Table 2 we give numerical results for R ⋆b for the square and hexagonal lattices. These results show that R ⋆b is largest for the hexagonal lattice. For these two simple lattices, in Table 2 we also give numerical results for R p ,defined by (3.6), as obtained from the explicit formula in Theorem 1 of [ ] and § ]. In Theorem 2 of [ ] it wasproved that, within the class of oblique Bravais lattices with unit area of the primitive cell, R p is minimized fora hexagonal lattice. Finally, in the fourth and fifth columns of Table 2 we give numerical results for K s and K gm ,as defined in Principal Results 3.2 and 4.2. Of the two lattices, we conclude that K s and K gm are largest for thehexagonal lattice. In addition, since R ⋆b is maximized and R p is minimized for a hexagonal lattice, it follows that K gs in Principal Result 5.1 is also largest for a hexagonal lattice. Thus, with respect to the two simple lattices, weconclude that the optimal stability thresholds in Principal Results 3.2, 4.2, and 5.1, occur for a hexagonal lattice.To show that the same conclusion regarding the optimal stability thresholds occurs for the class of oblique lattices,we need only show that R ⋆b is still maximized for the hexagonal lattice. This is done numerically below.We first consider lattices for which | lll | = | lll | . For this subclass of lattices, the lattice vectors are lll = (1 / p sin( θ ) , lll = (cos( θ ) / p sin( θ ) , p sin( θ )). In our computations, we first use a coarse grid to find an approximate locationin kkk -space of the minimum of R (0) and then we refine the search. After establishing by a coarse discretization thatthe minimum arises near a vertex of the adjoint lattice, we then sample more finely near this vertex. The finestmesh has a resolution of π/ R ⋆b we interpolate a paraboloid through the approximateminimum and the four neighbouring points and evaluate the minimum of the paraboloid. As we vary the lattice byincreasing θ , we use the approximate location of the previous minimum as an initial guess. The value of θ is increasedby increments of 0 .
01. Our numerical results in Fig. 5 show that the optimum lattice where R ⋆b ≡ min kkk R (0) ismaximized occurs for the hexagonal lattice where θ = π/
3. In Fig. 6 we also plot R p versus θ (cf. Theorem 1 of [ ]),6 D. Iron, J. Rumsey, M. J. Ward, J. Wei − . − . − . − . − . − . − .
075 0 . . . . . . . . . . R ⋆b θ − . − . − . − . − . − . − . − . .
90 0 .
95 1 .
00 1 .
05 1 .
10 1 .
15 1 . R ⋆b θ Figure 5. Minimum value R ⋆b of R b ( kkk ) for all oblique lattices of unit area for which lll = (1 / p sin( θ ) ,
0) and lll = (cos( θ ) / p sin( θ ) , p sin( θ )), so that | lll | = | lll | and | Ω | = 1. The vertical line denotes the hexagonal lattice forwhich θ = π/
3. Left figure: the angle θ between the lattice vectors ranges over 0 . < θ < .
7. Right figure: enlargementof the left figure near θ = π/
3. The vertical line again denotes the hexagonal lattice. − . − . − . − . − . − .
190 0 . . . . . . . . . . . R p θ Figure 6. Plot of the the regular part R p , as given in (6.24) (cf. [ ]), of the periodic source-neutral Green’s functionfor all oblique lattices of unit area for which lll = (1 / p sin( θ ) ,
0) and lll = (cos( θ ) / p sin( θ ) , p sin( θ )), so that | lll | = | lll | and | Ω | = 1. The vertical line denotes the hexagonal lattice for which θ = π/
3. The minimum occurs for the hexagon.given by(6.24) R p = − π log(2 π ) − π ln (cid:12)(cid:12)(cid:12) √ sin θ e ( ξ/ ∞ Y n =1 (1 − e ( nξ )) (cid:12)(cid:12)(cid:12) , e ( z ) ≡ e πiz , ξ = e iθ . Finally, we consider a more general sweep through the class of oblique Bravais lattices. We let lll = ( a,
0) and lll = ( b, c ), so that with unit area of the primitive cell, we have ac = 1 and b = a − cot θ , where θ is the angle between lll and lll . We introduce a parameter α by a = (sin θ ) α so that(6.25) c = (sin θ ) − α and b = cos θ (sin θ ) − α − . Then | lll | = | lll | when α = − / | lll | = 1 (which is independent of θ ) when α = 0, and | lll | = 1 when α = −
1. Inthe left panel of Fig. 7, we plot R ⋆b versus θ for α = − . , − . , − . , − . , − . ,
0. The angle, θ , at which the maximumoccurs, increases from π/ α = − . .
107 = π/ .
06 for α = 0. However, the value of the maximumis largest for α = − . α increases to zero. The regular hexagon occurs only at α = − . θ = π/
3. The vertical line in the plot is at θ = π/
3. Similarly, in the right panel of Fig. 7 we plot R ⋆b versus θ for he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems α = − . , − . , − . , − . , − . , − .
0. Since there is no preferred angular orientation for the lattice and since the scale isarbitrary, the plot is identical to the previous plot, in the sense that the curves for α = − . α = − . R ⋆b . These computational results lead tothe following conjecture: -0.09-0.088-0.086-0.084-0.082-0.08-0.078 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 P S f r ag r e p l a ce m e n t s R ⋆ b θα = 0 α = − . α = − . α = − . α = − . α = − . -0.09-0.088-0.086-0.084-0.082-0.08-0.078 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 P S f r ag r e p l a ce m e n t s R ⋆ b θα = 1 α = − . α = − . α = − . α = − . α = − . R ⋆b versus θ for oblique lattices with lll = ( a,
0) and lll = ( b, c ), where a = (sin θ ) α with b and c given in (6.25). Left panel: plots are for α = − . , − . , − . , − . , − . ,
0. Right panel: plots are for α = − . , − . , − . , − . , − . , − . Conjecture 6.1
Within the class of Bravais lattices of a common area, R ⋆b is maximized for a regular hexagonallattice. We have studied the linear stability of steady-state periodic patterns of localized spots for the GM and SchnakenburgRD models when the spots are centered for ε → | Ω | . Toleading order in ν = − / log ε , the linearization of the steady-state periodic spot pattern has a zero eigenvalue when D = D /ν for some D independent of the lattice and the Bloch wavevector kkk . The critical value D can be identifiedfrom the leading-order NLEP theory of [ ] and [ ]. This zero eigenvalue corresponds to a competition instabilityof the spot amplitudes (cf. [ ], [ ], [ ], and [ ]). By using a combination of the method of matched asymptoticexpansions, Floquet-Bloch theory, and the rigorous imposition of solvability conditions for perturbations of certainnonlocal eigenvalue problems, we have explicitly determined the continuous band of spectrum that lies within an O ( ν ) neighborhood of the origin in the spectral plane when D = D /ν + D , where D = O (1) is a de-tuningparameter. This continuous band is real-valued, and depends on the regular part of the Bloch Green’s function and D . In this way, for each RD model, we have derived a specific objective function that must be maximized in orderto determine the specific periodic arrangement of localized spots that is linearly stable for the largest value of D .A simple alternative method to derive this objective function was also given and applied to the GS model. From anumerical computation, based on an Ewald-type algorithm, of the regular part of the Bloch Green’s function thatdefines the objective function, we have shown within the class of oblique Bravais lattices that a hexagonal latticearrangement of spots is the most stable to competition instabilities.Although we have focused our analysis only on the Schnakenburg, GM, and GS models, our asymptotic method-ology to derive the model-dependent objective function that determines the optimally stable lattice arrangement of8 D. Iron, J. Rumsey, M. J. Ward, J. Wei spots is readily extended to general RD systems in the semi-strong interaction regime, such as the Brusselator RDmodel (cf. [ ]). Either the simple method of §
5, or the more elaborate but systematic method of § §
4, canthen be used to derive the objective function.There are a few open problems that warrant further investigation. One central issue is to place our formal asymp-totic theory on a more rigorous footing. In this direction, it is an open problem to rigorously characterize thecontinuous band of spectrum that lies near the origin when D is near the critical value. In addition, is it possible toanalytically prove Conjecture 6.1 that, within the class of oblique Bravais lattices of a common area, R ⋆b is maximizedfor a hexagonal lattice?As possible extensions to this work, it would be interesting to characterize lattice arrangements of spots thatmaximize the Hopf bifurcation threshold in τ . To analyze this problem, one would have to calculate any continuousband of spectra that lies within an O ( ν ) neighborhood of the Hopf bifurcation frequency λ = iλ I when τ − τ I ≪ τ I and λ I is the Hopf bifurcation threshold and frequency, respectively, on the Wigner-Seitz cell.We remark that we have not analyzed any weak instabilities due to eigenvalues of order λ = O ( ε ) associated withthe translation modes. It would be interesting to determine steady-state lattice arrangements of localized spots thatoptimize the linear stability properties of these modes. For these translation modes we might expect, in contrastto what we found in this paper for competition instabilities (see Remark 3.1 and Lemma 2.2), that it is the long-wavelength instabilities with | kkk | ≪ ], [ ]).Finally, it would be interesting to examine the linear stability properties of a collection of N ≫ R . For the finite domain problem, weexpect that there are N discrete eigenvalues (counting multiplicity) that are asymptotically close to the origin in thespectral plane when D is close to a critical threshold. Research in this direction is in progress. Acknowledgements
D. I. and M. J. W. were supported by NSERC (Canada). Prof. Juncheng Wei was partially supported by an EarmarkedGrant from RGC of Hong Kong and by NSERC (Canada). M.J.W. is grateful to Prof. Edgar Knobloch (U.C. Berkeley)for his comments regarding the de-stabilizing mechanisms of periodic weakly-nonlinear Turing patterns on lattices.
Appendix A Schnakenburg Model: Expansion of the Core Problem
We outline the derivation of the results of Lemma 3.1, as given in § ], and those of Lemma 5.1. To motivatethe appropriate scaling for solutions U , V , and χ to (3.2) for S →
0. Upon writing U = U S − p , V = V S p , where U and V are O (1) as S →
0, we obtain that the V -equation in (3.2) is unchanged, but that the U equation becomes∆ ρ U = S p UV ; U ∼ S p log ρ + S p χ as ρ → ∞ . From equating powers of S after first applying the divergence theorem, we obtain that 2 p = p + 1, which yields p = 1. Then, to ensure that U = O (1), we must have χ = O ( S − p ). This shows that if S = S ν / where ν ≪
1, theappropriate scalings are V = O ( ν / ), U = O ( ν − / ), and χ = O ( ν − / ).With this basic scaling, we then proceed to calculate higher order terms in the expansion of the solution to the coreproblem by writing S = S ν / + S ν / + · · · and then determining the first two terms in the asymptotic solution he Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems U , V , and χ to (3.2) in terms of S and S . The appropriate expansion for these quantities is (see (6.2) of[ ])(A.1) V ∼ ν / ( V + νV + · · · ) , ( χ , U ) = ν − / [( χ , U ) + ν ( χ , U ) + · · · ] . Upon substituting (A.1) into (3.2), and collecting powers of ν , we obtain that U and V satisfy∆ ρ V − V + U V = 0 ; ∆ ρ U = 0 , ≤ ρ < ∞ ,V → , U → χ as ρ → ∞ ; V ′ (0) = U ′ (0) = 0 , (A.2)where ∆ ρ V ≡ V ′′ + ρ − V ′ . At next order, U and V satisfy∆ ρ V − V + 2 U V V = − U V ; ∆ ρ U = U V , ≤ ρ < ∞ ,V → , U → S log ρ + χ as ρ → ∞ ; V ′ (0) = U ′ (0) = 0 . (A.3)Then, at one higher order, we get that U satisfies(A.4) ∆ ρ U = U V + 2 U V V , ≤ ρ < ∞ ; U ∼ S log ρ + χ as ρ → ∞ ; U ′ (0) = 0 . The solution to (A.2) is simply U = χ and V = w/χ , where w ( ρ ) > ρ w − w + w = 0 with w (0) > w → ρ → ∞ . To determine χ in terms of S we apply the divergencetheorem to the U equation in (A.3) to obtain(A.5) S = ∞ Z U V ρ dρ = bχ , b ≡ ∞ Z ρw dρ . It is then convenient to decompose U and V in terms of new variables U p and V p by(A.6) U = χ + U p χ , V = − χ wχ + V p χ . Upon substituting U = χ , V = w/χ , (A.5), and (A.6) into (A.3), and by using ∆ ρ w − w + 2 w = w , we readilyobtain that U p and V p are the unique radially symmetric solutions of (3.7 c ). Finally, we use the divergence theoremon the U equation in (A.4) to determine χ in terms of S as S = ∞ Z (cid:0) U V V + U V (cid:1) ρ dρ = − χ χ ∞ Z w ρ dρ + 1 χ ∞ Z (cid:0) wV p + w U p (cid:1) ρ dρ . We then use ∆ ρ V p − V p = − w U p − wV p in the integral, as obtained from (3.7 c ), and we simplify the resultingexpression by using U = χ and V = w/χ . This yields S = − b − χ S + b − S R ∞ V p ρ dρ , which gives (3.7 d ) for χ . This completes the derivation of Lemma 3.1.To obtain the result in Lemma 5.1, we set S = S ν / and S = 0 in (3.7) to obtain(A.7) V ∼ SS (cid:18) wχ + S S (cid:18) − χ wχ + V p χ (cid:19)(cid:19) , U ∼ S S (cid:18) χ + S S (cid:18) χ + U p χ (cid:19)(cid:19) , χ ∼ S χ S + Sb ∞ Z V p ρ dρ , since χ = S b − R ∞ V p ρ dρ from (3.7 d ). Finally, since S χ = b from (3.7 d ), (A.7) reduces to (5.1) of Lemma 5.1. Appendix B Gierer-Meinhardt Model: Expansion of the Core Problem
We outline the derivation of the results of Lemma 4.1 and Lemma 5.2. To motivate the scalings for the solution U , V , and χ to (4.2) as S →
0, we write U = U S p , V = V S p , where U and V are O (1) as S →
0. We obtain that the0
D. Iron, J. Rumsey, M. J. Ward, J. Wei V -equation in (4.2) is unchanged, but that the U equation becomes∆ ρ U = − S p V ; U ∼ − S − p log ρ + S − p χ as ρ → ∞ . From equating powers of S after applying the divergence theorem it follows that p = 1 − p , which yields p = 1 / χ = O ( S / ) ensures that U = O (1). This shows that if S = S ν where ν ≪
1, the appropriate scalings arethat V , U , and χ are all O ( ν ). To obtain a two-term expansion for the solution to the core problem, as given inLemma 4.1, we expand S = S ν + S ν + · · · and we seek to determine the solution U , V , and χ to (4.2) in termsof S and S . The appropriate expansion for these quantities has the form(B.1) ( V , U , χ ) = ν ( V , U , χ ) + ν ( V , U , χ ) + ν ( V , U , χ ) + · · · . Upon substituting (B.1) into (4.2), and collecting powers of ν , we obtain that U and V satisfy∆ ρ V − V + V /U = 0 ; ∆ ρ U = 0 , ≤ ρ < ∞ ,V → , U → χ as ρ → ∞ ; V ′ (0) = U ′ (0) = 0 , (B.2)where ∆ ρ V ≡ V ′′ + ρ − V ′ . At next order, U and V satisfy∆ ρ V − V + 2 V U V = V U U ; ∆ ρ U = − V , ≤ ρ < ∞ ,V → , U → − S log ρ + χ as ρ → ∞ ; V ′ (0) = U ′ (0) = 0 . (B.3)Then, at one higher order, we get that U satisfies(B.4) ∆ ρ U = − V V , ≤ ρ < ∞ ; U ∼ − S log ρ + χ as ρ → ∞ ; U ′ (0) = 0 . The solution to (B.2) is simply U = χ and V = χ w , where w ( ρ ) > ρ w − w + w = 0. Next, by applying the divergence theorem to the U equation in (B.3) we obtain(B.5) S = ∞ Z ρV dρ = χ b , b ≡ ∞ Z ρw dρ . It is then convenient to decompose U and V in terms of new variables U p and V p by(B.6) U = χ + S U p , V = χ w + S V p . Upon substituting U = χ , V = χ w , (B.5), and (B.6) into (B.3), and by using ∆ ρ w − w + 2 w = w , we readilyobtain that U p and V p are the unique radially symmetric solutions of (4.8 c ). Finally, we use the divergence theoremon the U equation in (B.4) to obtain 2 χ χ b + 2 χ S R ∞ wV p ρ dρ = S , which readily yields (4.8 d ).To obtain the result in Lemma 5.2, we set S = S ν and S = 0 in (4.8), with χ = S /b from (4.8 d ), to get(B.7) V ∼ r SS χ w + SS ( χ w + S V p ) , U ∼ r SS χ + SS ( χ + S U p ) , χ ∼ r SS χ + SS χ , where χ = − S b − R ∞ wV p ρ dρ from (4.8 d ). Since S = bχ from (4.8 d ), (B.7) reduces to (5.5) of Lemma 5.2. References [1] M. Abramowitz, I. Stegun,
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