The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime

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1 Preprint The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime T. KOLOKOLNIKOV, M. J. WARD, and J. WEI Theodore Kolokolnikov; Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada, Michael Ward; Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T Z, Canada, Juncheng Wei, Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong. January 3rd, ) The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 8, Suppl. 8), pp ]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and twodimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to O) time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold K c is determined such that a pattern consisting of K hot-spots is unstable to a competition instability if K > K c. This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reactiondiffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur. Key words: singular perturbations, hot-spots, reaction-diffusion, crime, nonlocal eigenvalue problem, Hopf Bifurcation. Introduction Recently, Short et. al. [9, 3, 3] introduced an agent-based model of urban crime that takes into account repeat or near-repeat victimization. In dimensionless form, the continuum limit of this agent-based model is the two-component reaction-diffusion PDE system A t = ε A A+PA+α, x Ω; n A =, x Ω,.a) τp t = D P PA ) A PA+γ α, x Ω; n P =, x Ω,.b) where the positive constants ε, D, α, γ and τ, are all assumed to be spatially independent. In this model, Px,t) represents the density of the criminals, Ax, t) represents the attractiveness of the environment to burglary or ) other criminal activity, and the chemotactic drift term D P A A represents the tendency of criminals to move towards sites with a higher attractiveness. In addition, α is the baseline attractiveness, while γ α)/τ represents the constant rate of re-introduction of criminals after a burglary. For further details on the model see [9]. In [9], the reaction-diffusion system.) with chemotactic drift term was derived from a continuum limit of a lattice-based model. It was then analyzed using linear stability theory to determine a parameter range for the

2 T. Kolokolnikov, M. J. Ward, J. Wei t=. t=3.8 5 t=5. 5 t=3.8 5 t=63.3 t=846. t=878. t= t=. t=8.7 t= t= t=4.9 5 t= t=5..5 t=. t=.6 t=. t=3.5 t=33.4 t=38. t=48.6 a) b) c) t=9. Figure. Numerical solution of.) at different times, for initial data close to a spatially homogeneous steady-state. Plots of Ax,t) are shown at the values of t indicated. a) One dimensional domain with parameter values α =, γ =, ε =., τ =, D =. Initial conditions are Px,) = α/γ and Ax,) = γ.cos6πx)). Turing instability leads to a formation of three hot-spots; one is annihilated almost immediately due to a fast-time instability, while the second hot-spot is annihilated after a long time. b) D =.5 with all other parameters as in a). Two hot-spots remain stable. c) Numerical solution of.) in a two-dimensional square of width 4. Parameters are α =, γ =, ε =.8, τ =, D =. Initial conditions are Px,) = α/γ and Ax,) = γ+rand.) where rand generates a random number between and. existence of a Turing instability of the spatially uniform steady-state. A weakly nonlinear theory, based on a multiscale expansion valid near the Turing bifurcation point, was developed in [3, 3] for.) for both one and twodimensional domains. This theoretical framework is very useful to explore the origins of various patterns that are observed in full numerical solutions of the model. However, the major drawback of a weakly nonlinear theory is that the parameters must be tuned near the bifurcation point of the Turing instability. When the parameters values are at an O) distance from the bifurcation point, an instability of the spatially homogeneous steady-state often leads to patterns consisting of localized structures. Such localized patterns for the crime model.), consisting of the concentration of criminal activity in localized spatial regions, are referred to as either hot-spot or spike-type patterns. A localized hot-spot solution, not amenable to an analytical description by a weakly nonlinear analysis, was observed in the full numerical solutions of [3]. As an illustration of localization behavior, in Fig. a) we plot the numerical solution to.) in the one-dimensional domain Ω = [,] with parameter values α =, γ =, ε =., τ =, and D =. The initial conditions, consisting of a small mode-three perturbation of the spatially homogeneous steady-state A e = γ and P e = γ α)/γ are first amplified due to linear instability. Shortly thereafter, nonlinear effects become significant and the solution quickly becomes localized leading to the formation of three hot-spots, as shown at t 4. Subsequently, one of the hot-spots appears to be unstable and is quickly annihilated. The remaining two hot-spots drift towards each other over a long

3 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime 3 time, until finally around t 8,, another hot-spot is annihilated. The lone remaining hot-spot then drifts towards the center of the domain where it then remains. Next, in Fig. b) we re-run the simulation when D is decreased to D =.5 with all other parameters the same as in Fig. a). For this value of D, we observe that the final state consists of two hot-spots. Similar complex dynamics of hot-spots in a two-dimensional domain are shown in Fig. c). It is the goal of this paper to give a detailed study of the existence and stability of steady-state localized hot-spot patterns for.) in both one and two-dimensional domains in the singularly perturbed limit ε D..) The assumption that ε D implies that the length-scale associated with the change in the attractiveness of potential burglary sites is much smaller than the length-scale over which criminals explore new territory to commit crime. In this limit, a singular perturbation methodology will be used to construct steady-state hot-spot solutions and to derive new nonlocal eigenvalue problems NLEP s) governing the stability of these solutions. From an analysis of the spectrum of these NLEP s, explicit stability thresholds in terms of D and τ for the initiation of O) time-scale instabilities of these patterns are obtained. In a one-dimensional domain, an additional stability threshold on D for the initiation of slow translational instabilities of the hot-spot pattern is derived. Among other results, we will be able to explain both the fast and slow instabilities of the localized hot-spots patterns as observed in Fig. a). In related contexts, there is now a rather large literature on the stability of spike-type patterns in two-component reaction-diffusion systems with no drift terms. The theory was first developed in a one-dimensional domain to analyze the stability of steady-state spike patterns for the Gierer-Meinhardt model cf. [, 3, 35, 37, 38, 34, 43, 46]) and, in a parallel development, the Gray-Scott model cf. [4, 5, 5, 3, 4, 8, ]). The stability theory for these two models was extended to two-dimensional domains in [39, 4, 4, 44, 43, ]. Related studies for the Schnakenburg model are given in [, 36, 4]. The dynamics of quasi-equilibrium spike patterns is studied for one-dimensional domains in [9, 6, 7, 33, ], and in a multi-dimensional context in [3, 4, 6, ]. More recently, in [9] the stability of spikes was analyzed for a reaction-diffusion model of species segregation with cross-diffusion. A common feature in all of these studies, is that an analysis of the spectrum of various classes of NLEP s is central for determining the stability properties of localized patterns. A survey of NLEP theory is given in [46], and in, a broader context, a survey of phenomena and results for far-from-equilibrium patterns is given in [5]. In contrast, for reaction-diffusion systems with chemotactic drift terms, such as the crime model.), there are only a few studies of the existence and stability of spike solutions. These previous studies have focused mainly on variants of the well-known Keller-Segel model cf. [8,, 8, 3]). We now summarize and illustrate our main results. In. we construct a multi hot-spot steady state solution to.) on a one-dimensional interval of length S. We refer to a symmetric hot-spot steady-state solution as one for which the hot-spots are equally spaced and, correspondingly, each hot-spot has the same amplitude. In. asymmetric steady-state hot-spot solutions, characterized by unevenly spaced hot-spots, are shown to bifurcate from the symmetric branch of hot-spot solutions at a critical value of D. In 3 we study the stability of steady-state K-hot-spot solutions on an interval of length S when τ = O). A singular perturbation approach is used to derive a NLEP that determines the stability of these hot-spot patterns to O) time-scale instabilities. In contrast to the NLEP s arising in the study of spike stability for the Gierer-Meinhardt model cf. [35]), this NLEP is explicitly solvable. In this way, a critical threshold K c+ is determined such that a

4 4 T. Kolokolnikov, M. J. Ward, J. Wei pattern consisting of K hot-spots with K > is unstable to a competition instability if and only if K > K c+. This instability, which develops on an O) time scale as ε, is due to a positive real eigenvalue, and it triggers the collapse of some of the hot-spots in the pattern. This critical threshold K c+ > is the unique root of see Principal Result 3. below) K+cosπ/K)) /4 = ) ) /4 S γ α) 3/4..3) D πεα In addition, from the location of the bifurcation point associated with the birth of an asymmetric hot-spot equilibrium, a further threshold K c is derived that predicts that a K-hot-spot steady-state with K > is stable with respect to slow translational instabilities of the hot-spot locations if and only if K < K c. This threshold is given explicitly by see 3.) below) K c = S )D /4γ α)3/4..4) πεα Since K c < K c+, the stability properties of a K-hot-spot steady-state solution with K > and τ = O) are as follows: stability when K < K c ; stability with respect to O) time-scale instabilities but unstable with respect to slow translation instabilities when K c < K < K c+ ; a fast O) time-scale instability dominates when K > K c+. As an illustration of these results consider again Fig. a). From the parameter values in the figure caption we compute from.3) and.4) that K c+.73 and K c.995. Therefore, we predict that the three hot-spots that form at t = 3.8 are unstable on an O) time-scale. This is confirmed by the numerical results shown at times t = 5 and t = 3.8 in Fig. a). We then predict from the threshold K c that the two-hot-spot solution will become unstable on a very long time interval. This is also confirmed by the full numerical solutions shown in Fig. a). In contrast, if we decrease D to D =.5 as in Fig. b) then we calculate from.3) and.4) that K c+.6 and K c.37. Our prediction is that the three hot-spot solution that emerges from initial data will be unstable on an O) time-scale, but that a two-hot-spot steady-state will be stable. These predictions are again corroborated by the full numerical results. In 4 we examine oscillatory instabilities of the amplitudes of the hot-spots in terms of the bifurcation parameter τ in.). From an analysis of a new NLEP with two separate nonlocal terms, we show that an oscillatory instability of the hot-spot amplitudes as a result of a Hopf bifurcation is not possible on the regime τ Oε ). This non-existence result for a Hopf bifurcation is in contrast to the results obtained in [35] for the Gierer-Meinhardt model showing the existence of oscillatory instabilities of the spike amplitudes in a rather wide parameter regime. However, for the asymptotically larger range of τ with τ = Oε ), in 4. we study oscillatory instabilities of a single hot-spot in the simplified system corresponding to letting D in.). In this shadow system limit, we show for a domain of length one that low frequency oscillations of the spot amplitude due to a Hopf bifurcation will occur when τ > τ c where τ c.39759γ α) 3 α ε. In 5 we extend our results to two dimensional domains. We first construct a quasi-equilibrium multi hot-spot pattern, and then derive an NLEP governing O) time-scale instabilities of the spot pattern. As in the analyses of [39, 4, 4, 4, 43, 44] for the Gierer-Meinhardt and Gray-Scott models, our existence and stability theory for localized hot-spot solutions is accurate only to leading-order in powers of /logε. In 5., we show from an analysis

5 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime 5 of a certain NLEP problem that for τ = O) a K-hot-spot solution with K > is unstable when K > K c where K c D /3 Ω γ α)α /3 [4π R w 3 dy ) ] /3 ε 4/3 logε) /3.737) D /3 α /3 γ α) Ω ε 4/3 logε) /3,.5) as ε. Here w is the radially symmetric ground-state solution of w w + w 3 = in R and Ω is the area of Ω. As an example, consider the parameter values as in Fig. c), for which Ω = 6. Then, from.5) we get K c Starting with random initial conditions, we observe from Fig. c) that at t = 9 we have K = 7.5 hot-spots, where we count boundary spots having weight / and corner spots having weight /4. Since K < K c, this is in agreement with the stability theory. Finally, in 5, we contrast results for Turing instabilities and Turing patterns with our results for localized hot-spots. We also propose a few open problems. Asymptotic Analysis of Steady-State Hot-Spot Solutions in -D In the -D interval x [ l,l], the reaction-diffusion system.) is A t = ε A xx A+PA+α τp t = D P x P ) A A x PA+γ α, x.a).b) with Neumann boundary conditions P x ±l,t) = A x ±l,t) =. Since P x P A A x = P/A ) x A, it is convenient to introduce the new variable V defined by so that.) transforms to V = P/A,.) A t = ε A xx A+VA 3 +α,.3a) τ A V ) t = D A V x )x VA3 +γ α..3b) To motivate the ε-dependent re-scaling of V that facilitates the analysis below, we suppose that D l and we integrate the steady-state of.3b) over l < x < l to obtain that V = c/ l l A3 dx, where c is some O) constant as ε. Therefore, if A = Oε p ) in the inner hot-spot region of spatial extent Oε), we conclude that l l A3 dx = Oε 3p ), so that V = Oε 3p ). In addition, from the steady-state of.3a), we conclude that in the inner region near a hot-spot centered at x = x we must have A + A 3 V = OA yy ), where y = x x )/ε and A = Oε p ). This implies that 3p + 3p ) = p, so that p =. Therefore, for D l we conclude that V = Oε ) globally on l < x < l, while A = Oε ) in the inner region near a hot-spot. Finally, in the outer region we must have A = O), so that from the steady-state of.3b), we conclude that D A V x α γ = O). Since )x V = Oε ), this balance requires that D = Oε ). Since V = Oε ) globally, while A = Oε ) in the core of a hot-spot, we conclude that within a hot-spot of criminal activity the density P = VA of criminals is O). In summary, this simple scaling analysis motivates the introduction of new O) variables v and D defined by V = ε v, D = D /ε..4)

6 6 T. Kolokolnikov, M. J. Ward, J. Wei In terms of.4),.3) transforms to A t = ε A xx A+ε va 3 +α, l < x < l; A x ±l,t) =,.5a) τε A v ) t = D A ) v x x ε va 3 +γ α, l < x < l; v x ±l,t) =..5b). A Single Steady-State Hot-Spot Solution We will nowconstructasteady-statehot-spotsolutiononthe interval l < x < l with apeakatthe origin.in orderto construct a K-hot-spot pattern on a domain of length S, with evenly spaced spots, we need only set l = S/K) and perform a periodic extension of the results obtained below on the basic interval l < x < l. As such, the fundamental problem considered below is to asymptotically construct a one-hot-spot steady-state solution on l < x < l. In the inner region, near the center of the hot-spot at x =, we expand A and v as A = A ε +A +, v = v +εv +, y = x/ε..6) From.5a) we obtain, in terms of y, that A j y) for j =, satisfy In contrast, from.5b), we obtain that v j for j =, satisfy A A +v A 3 =, < y <,.7a) A A ++3A A v = α v A 3, < y <..7b) A v ) =, A v +A A ) v =, < y <..8) In order to match to an outer solution, we require that v and v are bounded as y. In this way, we then obtain that v and v must both be constants, independent of y. We look for a solution to.7) for which the hot-spot has a maximum at y =. The homoclinic solution to.7a) with A ) = is written as where w is the unique solution to the ground-state problem A y) = v / wy),.9) w w+w 3 =, < y < ; w) >, w ) = ; w as y,.) given explicitly by w = sechy. Next, we decompose the solution A to.7b) as where w y) satisfies with w ) = and w as y. A = α v v 3/ w 3αw, L w w w +3w w = w, < y <,.) A key property of the operator L, which relies on the cubic exponent in.), is the remarkable identity that L w = 3w..) The proof of this identity is a straightforward manipulation of.) and the operator L in.). This property

7 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime 7 plays an important role in an explicit analysis of the spectral problem in 3. Here this identity is used to provide an explicit solution to.) in the form w = w /3. In this way, in the inner region the two-term expansion for A in terms of the unknown constants v and v is Ay) ε A y)+a y)+, A y) = ε v / wy), A y) = α [wy)] ) v v 3/ wy)..3) In the outer region, defined for ε x l, we have that v = O) and that A = O). From.5), we obtain that A = α+o), v = h x)+o), where from.5b), h x) satisfies h xx = ζ α γ) D α <, < x l; h x ±l) =, subject to the matching condition that h v as x ±. The solution to this problem gives the outer expansion v h x) = ζ [ l x ) l ] +v, < x l..4) Next, we must calculate the constantsv and v appearingin.3)and.4).we integrate.5b) over l < x < l and use v x = at x = ±l to get l ε va 3 dx = lα γ). l Since A = Oε ) in the inner region, while A = O) in the outer region, the dominant contribution to the integral in.) arises from the inner region where x = Oε). If we use the inner expansion A = ε A + A + o) from.3), and change variables to y = ε x, we obtain from.) that ) v A 3 dy +ε 3v A A dy +v A 3 +Oε ) = lα γ)..5) In.5), we emphasize that the first two terms on the left-hand side arise solely from the inner expansion, whereas the Oε ) term would be obtained from both the inner and outer expansions. By equating coefficients of ε in.5), we obtain that v = lγ α)/ A 3 dy, v = 3v A A dy/ A 3 dy, Then, upon using.3) for A and A, together with w = sechy, w3 dy = π, and w4 dy = 6/3, we readily derive from.) that v = π l α γ), v = 6v 3/ α w w 4) ) dy w3 dy We summarize our result for a single steady-state hot-spot solution as follows: = 4 π v3/ α = απ l 3 γ α) 3..6) Principal Result.: Let ε, and consider a one-hot-spot solution centered at the origin for.5) on the interval x l. Then, in the inner region y = x/ε = O), we have ) Ay) = w ε +α + v π w w +o), v v +εv +..7) In addition, in the inner region, the leading-order steady-state criminal density P from.) is P w. Here w =

8 8 T. Kolokolnikov, M. J. Ward, J. Wei A x v x a) b) Figure. Steady-state solution in one spatial dimension. Parameter values are D =,ε =.5,α =,γ =, and x [,]. a) The solid line is the steady state solution Ax) of.5) computed by solving the boundary value problem numerically. The dashed line corresponds to the first-order composite approximation given by.9) b) The solid line is the steady state solution for vx). Note the flat knee region obtained from the full numerical solution in the inner region near the center of the hot-spot. The dashed line is the leading-order asymptotic result.8). ε A) num) A) asy) A) asy) v) num) v) asy) v) asy) Table. Comparison of numerical and asymptotic results for the amplitude A max A) and for v) of a one-hotspot solution on [,] with D =, γ =, and α =. The -term and -term asymptotic results for A max and v) are obtained from.7). wy) = sechy is the homoclinic of.), while v and v are given in.6). In the outer region, Oε) < x l, then A α+o); v ζ l x ) l ) +v +o), ζ α γ) D α <..8) Note that to get a solution for A which is uniformly valid in both inner and outer region, we can combine the formulas.7) and.8). The resulting first-order composite solution is given explicitly by ) lγ α) x A α sech +α..9) πε ε) For a specific parameter set, a comparison of the full numerical steady-state solution of.5) with the composite asymptotic solution.9) is shown in Fig.. A comparison of numerical and asymptotic values for A) and v) at various ε is shown in Table. From this table we note that the two-term asymptotic expansion for v) agrees very favorably with full numerical results.

9 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime 9. Asymmetric Steady-State K-Hot-Spot Solutions Inthelimit ε,wenowconstructanasymmetricsteady-statek-hot-spotsolutionto.5)intheformofasequence of hot-spots of different heights. This construction will be used to characterize the stability of symmetric steady-state K-hot-spot solutions with respect to the small eigenvalues λ = o) in the spectrum of the linearization. Since the asymmetric solution is shown to bifurcate from the symmetric branch, the point of the bifurcation corresponds to a zero eigenvalue crossing along the symmetric branch. To determine this bifurcation point, we compute vl) for the one-hot-spot steady-state solution to.5) on x l, where l > is a parameter. This canonical problem is shown to have two different solutions. A K-hot-spot asymmetric solution to.5) is then obtained by using translates of these two local solutions in such a way to ensure that the resulting solution is C continuous. Since the details of the construction of the asymmetric solution is very similar to that in [36] for the Schnakenburg model, we will only give a brief outline of the analysis. The key quantity of interest is the critical value DK s of D for which an asymmetric K-hot-spot solution branch bifurcates off of the symmetric branch. To this end, we first calculate from.4) that vl) = γ α) α D Bl/q), D π α ) /4 q γ α) 3,.a) where the function Bz) on < z < is defined by Bz) z +/z..b) The function Bz) > in.b) has a unique global minimum point at z = z c =, and it satisfies B z) < on [,z c ) and B z) > on z c, ). Therefore, given any z,z c ), there exists a unique point z z c, ) such that Bz) = B z). This shows that given any l, with l/q < z c =, there exists a unique l, with l/q z > z c =, such that vl) = v l). We refer to solutions of length l and l as A-type and B-type hot-spots. Now consider the interval x [a,b] with length S b a. To construct a K-hot-spot steady-state solution to.5) on this interval with K hot-spots of type A and K = K K hot-spots of type B, arranged in any order across the interval, we must solve the coupled system K l +K l = S and Bl/q) = B l/q) for l l. Such solution exists only if l/q < zc and l/q > z c with z c =. The bifurcation point corresponds to the minimum point where l = l = q. With D = D /ε, this yields that D π α ) /4 Dπ l = γ α) 3 = ε / α ) /4 γ α) 3..) At this value of the parameters, a steady-state K-hot-spot asymmetric solution branch bifurcates off of the symmetric K-hot-spot branch. This critical value of D determines the small eigenvalue stability threshold in the linearization of the symmetric K-hot-spot steady-state solution. For a symmetric configuration of K hot-spots on an interval of length S we have Kl = S so that the critical value D = DK S, as defined by.), can be written as D S K = γ α)3 π α ) 4 S..) K A more detailed construction of the asymmetric solution branches parallels that done in [36] for the Schnakenburg model and is left to the reader.

10 T. Kolokolnikov, M. J. Ward, J. Wei 3 The NLEP Stability of Steady-State -D Hot-Spot Patterns We now study the stability of the K-hot-spot steady-state solution to.5) that was constructed in. The analysis for the large O) eigenvalues in the spectrum of the linearization is done in several distinct steps. Firstly, we let A e, v e denote the one-hot-spot quasi-steady-state solution to.5) on the basic interval x l, which was given in Principal Result.. Upon introducing the perturbation A = A e +φe λt, v = v e +ψe λt, 3.) we obtain from the linearization of.5) that ε φ xx φ+3ε v e A eφ+ε 3 A 3 eψ = λφ, D εa e ψ x +A e v ex φ ) x 3ε A e v eφ ε 3 ψa 3 e = τλε εa e ψ +A ev e φ ). 3.a) 3.b) We consider 3.a) and 3.b) on x l subject to the Floquet-type boundary conditions where z is a complex parameter. φl) = zφ l), φ l) = zφ l), ψl) = zψ l), ψ l) = zψ l), 3.c) For simplicity, in this section we will set τ = in 3.b). The analysis of the possibility of Hopf bifurcations induced by taking τ is studied in 4. After formulating the NLEP associated with solving 3.) for arbitrary z, we then must determine z so that we have the required NLEP problem for a K-hot-spot pattern on [ l,k )l] with periodic boundary conditions. This is done by translating φ and ψ from the interval [ l,l] to the extended interval [ l,k )l] in such a way that the extended φ and ψ have continuous derivatives at x = l,3l,...,k 3)l. It follows that φ[k )l] = z K φ l), and hence to obtain periodic boundary conditions on an interval of length Kl we require that z K =, so that z j = e πij/k, j =,...,K. 3.3) By using these values of z j in the NLEP problem associated with 3.), we obtain the stability threshold of a K- hot-spot solution on a domain of length Kl subject to periodic boundary conditions. The last step in the analysis is then to extract the stability thresholds for the corresponding Neumann problem from the thresholds for the periodic problem, and to choose l appropriately so that the Neumann problem is posed on [,]. This is done below. This Floquet-based approach to determine the NLEP problem of a K-hot-spot steady-state solution for the Neumann problem has been used previously for reaction-diffusion systems exhibiting mesa patterns [], for the Gierer-Meinhardt model [34], and for a cross-diffusion system [9]. We now implement the details of this calculation. The asymptotic analysis for ε of 3.) proceeds as follows. In the inner region with y = ε x, we use A e = Oε ) and v ex, to obtain from 3.b) that to leading order [ ] w ψ y = in the inner region, where w is the homoclinic satisfying.). To prevent exponential growth for ψ y as y, we must take ψ = ψ where ψ is a constant to be determined. Then, for 3.a) we look for a localized inner eigenfunction in the form φ Φy), y = ε x. Upon using the leading-orderapproximationa e ε v / w in 3.a), we obtain to leading order that Φy) satisfies L Φ+ w 3 ψ = λφ, < y < ; L Φ Φ Φ+3w Φ, 3.4) v 3/

11 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime with Φ as y. Here v is given in.6). In the outer region, away from the hot-spot centered at x =, we have A e α and v = O), so that 3.a) yields φ = ε3 α 3 ψ. 3.5) λ+ Then, from 3.b), together with A e α, we obtain the outer approximation D [ εα ψ x +Oε 3 ) ] x = Oε3 ), which yields the leading-order outer problem ψ xx =, < x l, 3.6) subject to the Floquet-type boundary conditions 3. c). The matching condition for the inner and outer representations of ψ is that lim x ψx) = ψ, where ψ is the unknown constant required in the spectral problem 3.4). However, the problem for ψ is not yet complete, as it must be supplemented by appropriate jump conditions for ψ x across x =. We now proceed to derive this jump condition. We first define an intermediate scale η satisfying ε η, and we integrate 3.b) over x η to get D εa e ψ x +A e v ex φ ) η η = η η 3ε A ev e φ+ε 3 ψa 3 ) e dx. 3.7) We use the limiting behavior as x ± of the outer expansion to calculate the terms on the left hand-side of 3.7). From A e α,.4) to calculate v ex ± ), and 3.5) to calculate φ ± ), we obtain that D εa e ψ x ) η η D εα ψ x + ) ψ x ) ) = εd α [ψ x ], 3.8a) D A e v ex φ) η η D α [ φ + )v ex + ) φ )v ex ) ] = 4D αφ + )v ex + ) 4ε3 α ψ)γ α)l. 3.8b) λ+ Here we have defined [ψ x ] ψ x + ) ψ x ). Next, since η Oε), we can estimate the integrals on the right-hand side of 3.7) by their contributions from the inner approximation A e ε v / wy), ψ ψ, φ Φy), and v e v. In this way, we calculate η η 3ε A e v eφ+ε 3 ψa 3 e) dx 3ε w Φdy + εψ v 3/ w 3 dy. 3.9) Upon substituting 3.8) and 3.9) into 3.7), we obtain the following jump condition for ψ x across x = : D α [ψ x ] = ψ v 3/ w 3 dy 4ε α ) γ α)l +3 w Φdy. 3.) λ+ For the range λ >, we can neglect the negligible Oε ) term in the jump condition 3.). In this way, the problem for the outer eigenfunction ψx) is to solve ψ xx =, < x l; ψl) = zψ l), ψ l) = zψ l), 3.a) subject to the continuity condition ψ + ) = ψ ) = ψ and the following jump condition across x = : a [ψ x ] +a ψ) = a ; a D α, a = v 3/ w 3 dy, a = 3 Upon calculating ψ) = ψ from this problem, the NLEP is then obtained from 3.4). w Φdy. 3.b) Upon solving 3.) for ψx), and evaluating the result at x = we get )[ ) ] v 3/ ψ = 3 w Φdy D α v 3/ w3 dy l z ). 3.) w3 dy z

12 T. Kolokolnikov, M. J. Ward, J. Wei Next, we use.6) for v and w3 dy = π to simplify v 3/ ψ. In addition, we use 3.3) to calculate z ) = +Rez) = [ cosπj/k)], j =,...,K. 3.3) z Upon substituting these results into 3.4), we obtain the following NLEP for a K-hot-spot steady-state on a domain of length Kl subject to periodic boundary conditions: L Φ χ j w 3 w Φdy w3 dy = λφ, < y < ; Φ, y, 3.4a) χ j 3 [+ D α π ] 4l 4 γ α) 3 cosπj/k)), j =,...,K. 3.4b) The final step in the analysis is extract the NLEP for the Neumann problem from the NLEP 3.4) for the periodic problem. More specifically, the stability thresholds for a K-hot-spot pattern with Neumann boundary conditions can be obtained from the corresponding thresholds for a K-hot-spot pattern with periodic boundary conditions on a domain of twice the length. To see this, suppose that φ is a Neumann eigenfunction on the interval [,a]. Extend it by an even reflection about the origin to the interval [ a, a]. Such an extension then satisfies periodic boundary conditions on [ a, a]. Alternatively, if φx) is an eigenfunction with periodic boundary conditions at the edge of the interval [ a,a], then define ˆφx) = φx)+φ x). Then, ˆφ is a eigenfunction for the Neumann boundary problem on [,a]. Therefore, to obtain the NLEP problem governing the stability of an steady-state K-hot-spot pattern on an interval of length S subject to Neumann boundary conditions, we simply replace cosπj/k) with cosπj/k) in 3.4) and then set l = S/K) in the NLEP of 3.4). In this way, we obtain the following main result: Principal Result 3.: Consider a K-hot-spot solution to.5) on an interval of length S subject to Neumann boundary conditions. For ε, and τ = O), the stability of this solution with respect to the large eigenvalues λ = O) of the linearization is determined by the spectrum of the NLEP L Φ χ j w 3 w Φdy w3 dy χ j = 3 [ + D α π K 4 4γ α) 3 where wy) is the homoclinic solution satisfying w w+w 3 =. = λφ, < y < ; Φ, y, 3.5a) ) ] 4 cosπj/k)), j =,...,K, 3.5b) S The stability threshold for D is characterized by the largest possible value of D for which the point spectrum of 3.5) satisfies Reλ) < for each j =,...,K. In contrast to the typical NLEP problem associated with spike patterns in the Gierer-Meinhardt, Gray-Scott, and Schnakeneburg reaction-diffusion models studied in [3], [4], [5], [], [5], [35], [36], and [43], the point spectrum for the non-self-adjoint problem 3.5) is real, and can be determined analytically. This fact, as we now show, relies critically on the identity L w = 3w from.). Lemma 3.: Consider the NLEP problem L Φ cw 3 w Φdy = λφ, < y < ; Φ, y, 3.6) for an arbitrary constant c corresponding to eigenfunctions for which w Φdy. Consider the range Reλ) >

13 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime 3. Then, on this range there is only one element in the point spectrum, and it is given explicitly by λ = 3 c w 5 dy. 3.7) To prove this we consider only the region Reλ) >, where we can guarantee that Φ exponentially as y. The continuous spectrum for 3.6) is λ <, with λ real. To establish 3.7) we use Green s identity on w L Φ ΦL w ) dy =. Since L Φ = cw 3 w Φdy+λΦ and L w = 3w, w and Φ, which is written as this identity reduces to ) w Φdy λ 3+c w 5 dy =, from which the result 3.7) follows. We remark that for the corresponding local eigenvalue problem L Φ = νφ, it was proved in Proposition 5.6 of [3] that the point spectrum consists only of ν = 3 and the translation mode ν = with odd eigenfunction), and that there are no other point spectra in < ν <. When c =, we observe that 3.7) agrees with ν. As a further remark, the result 3.7), when extrapolated into the region λ <, suggests that there is a critical value of c for which the discrete eigenvalue bifurcates out of the continuous spectrum into the region λ > on the real axis. By applying Lemma 3. to the NLEP 3.5) we conclude that Reλ) < if and only if ) χ j < 3 w3 dy =, j =,...,K. 3.8) w5 dy In obtaining the last equality in 3.8) we calculated the integrals using w = sechy. Since χ = 3 < χ < χ <...χ K, a one-hot-spot solution is stable for all D, while the instability threshold for a multi hot-spot pattern is set by χ K. In this way, we obtain the following main stability result. Principal Result 3.: Consider a K-hot-spot solution to.5) on an interval of length S with K > subject to Neumann boundary conditions. For τ =, and in the limit ε, this solution is stable on an O) time-scale provided that D < D L K, where D L K γ α) 3 S/) 4 K 4 α π [+cosπ/k)]. 3.9) In terms of the original diffusivity D, given by D = ε D, the stability threshold is DK L = ε DK L when K >. Alternatively, a one-hot-spot solution is stable for all D >, provided that D is independent of ε. Although we have not calculated the stability threshold for the small eigenvalues for which λ as ε in the spectrum of the linearization 3.), we conjecture that this stability threshold is the same critical value D S K of D, given in.), for which an asymmetric K-hot-spot steady-state branch bifurcates off of the symmetric K-hot-spot branch. This simple approach to calculate the small eigenvalue stability threshold, which avoids the lengthy matrix manipulations of [], has been validated for the Gierer-Meinhardt, Gray-Scott, and Schnakenburg reaction-diffusion models in [37], [36], and [8]. Since DK S < DL K for K, we conclude that a symmetric K-hot-spot steady-state solution is stable with respect to both the large and the small eigenvalues only when D < DK S. We make two remarks. Firstly, for the case of a single hot-spot where K = we expect that the stability threshold for D will be exponentially large in /ε, and similar to that derived in [4] for the Gierer-Meinhardt model in the near-shadow limit. Secondly, we remark that the possibility of stabilizing multiple hot-spots for.) is in direct contrast to the result obtained in the analysis of [3] of spike solutions for a Keller-Segel-type chemotaxis model with

14 4 T. Kolokolnikov, M. J. Ward, J. Wei Figure 3. Instabilities of a two-hot-spot steady-state solution induced by increasing D. Left: two hot-spots are stable with D = 5. Middle: two hot-spots exhibit a slow-time instability when D = 3. Right: there is a fast-time instability when D = 5. The parameter values are fixed at ε =.7 α =, and γ =, on the interval x [,3]. The initial condition for the full numerical solution of.3) consists of two hot-spots that are perturbed slightly from the steady-state locations. a logarithmic sensitivity function for the drift term. For this chemotaxis problem of [3], only a one-spike solution can be stable. 3. Numerical Results We now compare our stability predictions with results from full numerical solutions of.3). As derived above, under Neumann boundary conditions the thresholds on D for the stability of a symmetric K-hot-spot pattern on a domain of length KL are DK S L4 γ α) 3 ) ε α π, DK L = DK S +cosπ/k). 3.) To numerically validate these thresholds, we choose ε =.7, α =, γ =, L = and K =, so that we have an interval of length S = 4. For these parameters, our predicted stability thresholds are D S K.67 and DL K 4.33, and our initial condition is a two-hot-spot solution with hot-spot locations slightly perturbed from their steady-state values. For our full numerical solutions of.3) we choose either D = 5, D = 3, or D = 5. Our stability theory predicts the following; the two hot-spots are stable when D = 5; the two hot-spots are unstable with respect to only the small eigenvalues when D = 3; the two hot-spots are unstable with respect to both the small and large eigenvalues when D = 5. The full numerical results shown in Fig. 3 confirm this prediction from the asymptotic theory. 4 Hopf Bifurcation of K-Hot-Spot Steady-State Solutions In this section we study the spectrum of 3.) for τ >. This is done by first deriving an NLEP similar to 3.5). Since the analysis leading to the new NLEP is very similar to that in 3, we will only outline it here briefly. For τ Oε ), we get ψ ψ in the inner region x = Oε), and hence 3.4) for the inner approximation Φy) for φ remains valid. For τ Oε ), we get to leading-order that ψ xx = in the outer region < x l and so 3.6)

15 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime 5 still holds. However, for τ, the jump conditions 3.7) 3.9) must be modified. In place of 3.7), we get D εa e ψ x +A e v ex φ ) η η = η η 3ε A e v eφ+ε 3 ψa 3 ) η e dx+ε τλ εa e ψ +A e v e φ ) dx. 4.) η The left hand-side of 4.) was estimated in 3.8), while the first two terms on the right-hand side of 4.) were estimated in 3.9). We then use A e ε v / wy), ψ ψ, φ Φy), and v e v, to estimate the last term on the right hand-side of 4.) as ε τλ η η εa e ψ +A e v e φ ) dx ε τλ[ ψ v w dy + v ] wφdy. 4.) Upon substituting 3.8), 3.9), and 4.), into 4.), we obtain that [ D εα [ψ x ] +Oε 3 ) = ε 3 w Φdy + ψ ] [ w 3 dy +ε ψ τλ w dy + ] v V 3/ wφdy, 4.3) v which suggests the distinguished limit τ = Oε ). Upon defining τ = O) by τ = ε τ, 4.4) 4.3) yields the jump condition 3.b) for ψ across x =, where a, a, and a in 3.b) are to be replaced by a = Dα, a = v 3/ w 3 dy τ λ v w dy, a = 3 w Φdy + v τ λ With this modification of the coefficients in 3.b), the outer problem for ψ is still 3.). This problem is readily solved for ψx), and we obtain that ψ = ψ) is given by [ ) v 3/ ψ = 3 w Φdy +τ λ v wφdy )] [ D α π z ) w3 dy w3 dy 8l 4 γ α) 3 z wφdy. 4.5) + τ ] λ. 4.6) lγ α) Finally, upon substituting.6) and 3.3) into 4.6), the NLEP problem for the Floquet problem on [ l, l] follows from 3.4). As shown in 3, this problem allows us to readily determine the corresponding NLEP for the Neumann boundary condition problem on an interval of length S. The result is summarized as follows: Principal Result 4.: Let τ = Oε ) as ε and consider a steady-state k-hot-spot solution on an interval of length S with Neumann boundary conditions. Define τ c = O) by τ = ε Sγ α)τ c /4K). Then, the stability of a symmetric K-hot-spot steady-state solution is determined by the NLEP ) L Φ 3χ j w 3 w Φdy χ j w3 dy w3 wφdy = λφ, < y <, 4.7a) with Φ as y. Here, we have defined χ j, χ j, and β j by χ j [β j +τ c λ], χ j τ c λ)χ j, β j + D α π K 4 4γ α) 3 S ) 4 cosπj/k)), j =,...,K. 4.7b) This NLEP, with two separate nonlocal terms, is significantly different in form from the NLEP s derived for the Gierer-Meinhardt and Gray-Scott models studied in [3], [4], [5], [35], and [5]. Principal Result 4.: There is no value of τ c > for which the NLEP of 4.7) has a Hopf bifurcation. We note that there is a key step in the derivation of Principal Result 4. which relies on a numerical computation, see below. A completely computer-free derivation of this result is still an open problem.

16 6 T. Kolokolnikov, M. J. Ward, J. Wei [ w L ρµ) = w [ L +µ ) ] L wdy +µ) ] L wdy wl +µ) wdy w L +µ ) wdy µ Figure 4. A plot of the numerical result for ρµ), as obtained from 4.). Note that ρµ) is monotone increasing. Derivation of Principal Result 4.: Weusethenotation hdy hdy.uponusinggreen sidentity w LΦdy = ΦLw dy and L w = 3w, together with 4.7a), we obtain w 5 ) dy 3 3χ j w3 dy w Φdy χ j ) w 5 dy ) wφdy = λ w Φdy. Upon solving for w Φdy in terms of wφdy, and then substituting into 4.7a), we get L Φ fλ)w 3 wφdy = λφ, fλ) = We then simplify fλ) by using 4.7b), together with w 5 = 3/) w 3, to obtain [ 6χ j w a) χ j χ j 3 λ) w3 dy)] fλ) = β j τ c λ + 9 τ c λ3 λ). 4.8b) Next, we observe that the eigenvalues λ of the NLEP problem 4.8) are the roots of the transcendental equation fλ) = wl λ) w 3 dy. 4.9) Upon recalling that L w = w 3, we calculate wl λ) w 3 dy = wl λ) [L λ)w +λw] dy = w dy + λ wl λ) wdy. Substituting this result together with 4.8b) and w = 4 into 4.9), we obtain that λ is a root of λ wl λ) wdy = β j τ c λ 9 τ c λ3 λ). 4.) TodeterminewhetheraHopfbifurcationispossiblewesetλ = iλ I in4.)andreplacel λ) byl λ) = ) L +λ L I +iλ I ). Then, upon comparing the real and imaginary parts in the resulting expression, we obtain that µ λ I and any Hopf bifurcation threshold τ c must be the roots of the coupled system [ L w +µ ) ] L wdy = 4β j τ c µ + 54 τ c µ9+µ), w L +µ ) 8 wdy = τ c µ9+µ). 4.)

17 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime 7 Upon eliminating τ c from 4.), we obtain a transcendental equation solely for µ = λ I >, [ L ρµ) = 3 β j β w j +µ ) ] L wdy µ, where ρµ) 9 wl +µ). 4.) wdy By using the identities L w = w 3 and L w = w+yw )/, the limiting behavior for ρµ) is readily calculated as wl wdy ρ ) = = 8 wl w dy 3 ; ρ) = wdy L w) = 36 dy π Moreover, direct numerical computations of ρµ) show that it is an increasing function of µ see Fig. 4). On the other hand, for µ > we have 3 β j βj 9 µ < since β j. It follows that 4.) cannot have any solution with µ >. Consequently, there is no Hopf bifurcation on the parameter regime τ = Oε ). Such a non-existence result for Hopf bifurcations for the crime model when τ = Oε ) is qualitatively very different than for the Gierer-Meinhardt and Gray-Scott models, analyzed in [35] and [5], where Hopf bifurcations occur in wide parameter regimes. 4. A Hopf Bifurcation for the Shadow Limit Principal Result 4. has shown that there is no Hopf bifurcation for the regime τ = Oε ). However, a Hopf bifurcation can and does appear when τ = Oε ). As will be shown below, in such a regime the amplitude of the hot-spot becomes oscillatory with an asymptotically large temporal period, due to an eigenvalue that is dominated, to leading-order in ε, by its pure imaginary part. To illustrate this phenomenon, in this section we analytically derive the condition for a Hopf bifurcation of a single boundary spot on a domain of length one. To further simplify our computations,wewillassumethatd in.5b)is takensufficiently largesuchthat vx,t) = vt) canbe approximated by atime-dependent constant. The limit D is calledthe shadow-limitcf. [38]). Ourmain result is the following: Principal Result 4.3: Suppose that D and consider a half hot-spot of.5) located at the origin on the domain [,], as constructed in Principal Result.. Define τ c by ) 4 π γ α) 3 τ c = 36π α = γ α)3 α. 4.3) Let τ = τ /ε. Then, there is a Hopf bifurcation at τ = τ c. That is, the hot spot is stable for τ < τ c and is unstable for τ > τ c. Destabilization takes place via a Hopf bifurcation. More precisely, when τ = O), the related stability problem has an eigenvalue near the origin with the following asymptotic behavior as ε : { } { γ α α λ ± iε / π )} γ α) 4 π + ε. 4.4) τ γ α) τ 7 Numerical example. To illustrate Principal Result 4.3 we take γ = 4,α =,ε =.5 and D =. Then, 4.3) yields τ c Now take τ =.95 so that 4.4) yields the eigenvalue λ.48i.59. We then expect the single hot-spot to be stable, although it will exhibit long transient oscillations. From the eigenvalue, we can estimate the period of the oscillation to be P = π This agrees with full numerical solutions of 4.5) as shown in Fig. 5a). Next, we increase τ to.5, while keeping the other parameters the same. In this case, τ > τ c so that the hot-spot is unstable in the limit ε. However, τ =.5 is very close to the threshold value, and with ε =.5, we expect even longer transients with the final state still unclear at t = 3. This behavior is shown in Fig. 5b). Finally,asshowninFig.5c),whenweincreaseτ to.35,weclearlyobserveoscillationsofanincreasingamplitude.

18 8 T. Kolokolnikov, M. J. Ward, J. Wei a) b) c) Figure 5. maxu versus t with ε =.5,α =,γ = 4,τ = τ /ε with τ as given in the figure. a) τ < τ c =.74; damping is observed. b) τ > τ c but τ is very close to τ c; eventual fate of the oscillation is unclear. c) τ > τ c; oscillations of increasing amplitude are observed. Derivation of Principal Result 4.3: We begin by re-writing.5) as a shadow system. For convenience, we also rescale A as A = u/ε. In terms of this scaling, u = O) in the interior of the hot-spot. Expanding v in powers of D we then obtain to leading order that vx,t) vx). We then integrate.5b) and use the no-flux boundary conditions to obtain the following shadow-limit system on x [, ]: ) u t = ε u xx u+vu 3 +εα, τ v u dx = µ ε v u 3 dx; u x,t) = u x,t) =, 4.5a) where we have defined µ by t µ γ α. 4.5b) The shadow problem 4.5) is the starting point of our analysis. The corresponding steady-state system is = ε u xx u+vu 3 +εα, µ = ε v u 3 dx; u x,t) = u x,t) =. 4.6) As shown below, in order to analyze the Hopf bifurcation it is necessary to construct the steady-state solution to two orders in ε. To do so, we let y = x/ε and we expand u = u +εu..., v = v +εv ) By substituting this expansion into 4.6), and equating powers of ε, we obtain that u = v / w, where wy) is the positive homoclinic solution of w yy w+w 3 =. In addition, u satisfies L u = α w 3 v v 3/, 4.8) where the operator L is defined by L u u yy u+3w u. This operator has several key readily derivable identities, which allows us to determine the solution u to 4.8) as L ) = +3w, L w = w 3, L w = 3w, u = α αw v v 3/ w. 4.9) Next, to determine v and v we must calculate the integral in 4.6) for ε. This yields that µ = ε vu 3 dx /ε v +εv ) u 3 +3u u ε ) dy v / w 3 dy +ε 3u u v +v u 3 ) dy +Oε ).

19 The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime 9 By equating coefficients of ε, we get that v and v satisfy v / w 3 dy = µ, 3u w dy +v v 3/ w 3 dy =. Upon using the solution 4.9) for u, the unknown v can be determined in terms of a quadrature as v = 6αv 3/ w w 4) dy. w 3 dy The integrals defining v and v are then calculated explicitly by using w = sechy, which yields the explicit formulae w dy = 4, w 3 dy = wdy = π, w 4 dy = 6 3, v = π µ, v = απ µ 3. 4.) Next, we study the stability of this solution. For convenience, we extend the problem to the interval [,] by even reflection. We linearize 4.5) around the steady-state solution to obtain the eigenvalue problem λφ = ε φ φ+3u vφ+u 3 ψ; τλ /ε /ε φuv +u ψ ) dy = ε where the constant ψ denotes the perturbation in v. Upon solving for ψ we obtain /ε /ε 3u vφ+u 3 ψ ) dy, ε φ φ+3u vφ+u 3 ψ = λφ; ψ = τλε /ε /ε φuvdy + /ε /ε 3u vφdy τλε /ε /ε u dy + /ε. 4.) /ε u3 dy To motivate the analysis below, we first suppose that τλε. Then, by using u w/ v and v v, 4.) reduces to leading order to the NLEP L φ w 3 φw w = λφ. Here and below, f denotes fdy. This problem has a zero eigenvalue corresponding to the eigenfunction φ = w. All other discrete eigenvalues satisfy Reλ) < cf. [46]). Therefore, the critical eigenvalue will be a perturbation of the zero eigenvalue. A posteriori computations shows that the correct anzatz is in fact λ = ε / λ +ελ +... ; τ = τ ε. The analysis below shows that λ is purely imaginary, and hence determines the frequency of the oscillation, but not its stability. Therefore, a two-term expansion in λ must be obtained in order to determine the stability of the oscillations. As such, we must expand all quantities in the shadow problem up to Oε). The delicate part in the calculation is to note that /ε /ε u = where the last integral is in fact Oε) as a result of ε /ε /ε u = ε u +ε /ε /ε /ε u u +ε u, /ε α+...) = α ε+... Thus, this term jumps an order and is comparable in magnitude to ε u u. The remaining part of the analysis

20 T. Kolokolnikov, M. J. Ward, J. Wei is more straightforward. We let τ = τ ε and expand ψ in 4.) as so that the eigenvalue problem for φ from 4.) becomes ψ = τλε φuv + 3u vφ τλε u + u 3 = ψ +ε / ψ / +εψ, ε / λ +ελ )φ = L φ+3u v +6u u v )φε+u 3 ψ +ε / ψ / +εψ ) +3u u ψ ε = L φ+ε / L φ+εl 3 φ. 4.) After tedious but straightforward computations, the three operators in 4.) are given by wφ L φ L φ w w3 ; L φ ψ / u 3 = c wφ+c ) w φ w 3 ; 4.3a) 4.3b) L 3 φ 3u v +6u u v )φ+3u u ψ +u 3 ψ = c w+c 3 w +c 4 w 3) φ+ c 5 w +c 6 w 3 +c 7 w 4) wφ+c 8 w 3 φ+c 9 w 3 w φ, 4.3c) in terms of the coefficients c,...,c 9 defined by c = µ ; c = 3µ ; c = 3πα ; c 3 = ; c 4 = c ; c 5 = 3 πα 4τ λ 4πτ λ µ 4µ c 6 = µλ 4τ λ µ 8τ + π α λ 8µ ; c 7 = c 5 ; c 8 = π α πα 4µ ; c 9 = 4µ + 3µλ 3µ 4πτ λ + 8πτ. 4.4) λ Finally, we expand φ in 4.) as φ = w+ε / φ +εφ, and equate powers of ε in 4.). This yields the following problems for φ and φ : λ w = L φ +L w, 4.5) λ w+λ φ = L φ +L φ +L 3 w. 4.6) To determine λ and φ we must formulate the appropriate solvability condition based on the adjoint operator L of L defined by L φ L φ w 3 φ w. 4.7) w Since L admits a zero eigenvalue of multiplicity one, then so does L. In fact, w defined by w yw y +w)/, 4.8) is the unique element in the kernel of L, i.e. L w =, owing to the following two readily derived identities: w L w 3 w = w, =. 4.9) w Next, we impose a solvability condition on 4.5) in the usual way. We multiply 4.5) by w and integrate by parts to derive that λ = w L w w w = c w +c w 3 ) w w 3 w w = c w +c ) w 3, 4.3)