Instability thresholds and dynamics of mesa patterns in reaction-diffusion systems

Transcription

1 Instability thresholds and dynamics of mesa patterns in reaction-diffusion systems Rebecca McKay and Theodore Kolokolnikov epartment of Mathematics and Statistics, alhousie University, Canada Abstract We consider a class of one-dimensional reaction-diffusion systems, { ut = u xx fu, w. = w xx gu, w Under some generic conditions on the nonlinearities f, g, in the singular limit, and for a fixed independent of, such a system exhibits a steady state consisting of sharp back-to-back interfaces which is stable in time. On the other hand, it is also known that in the so-called shadow limit =, the periodic pattern having more than one interface is unstable. In this paper, we analyse in detail the transition between the stable patterns when = O and the shadow system when =. We show that this transition occurs when is exponentially large in and we derive instability thresholds 3... such that a periodic pattern with K interfaces is stable if < K and is unstable when > K. We also study the dynamics of the interfaces when is exponentially large; this allows us to describe in detail the mechanism leading to the instability. irect numerical computations of stability and dynamics are performed; excellent agreement with the asymptotic results is observed. Introduction One of the most prevalent phenomena observed in reaction-diffusion systems is the formation of mesa patterns. Such patterns consist of a sequence of highly localized interfaces or kinks that are separated in space by regions where the solution is nearly constant. These patterns has been studied intensively for the last three decades and by now an extensive literature exists on this topic. We refer for example to [,, 3, 4, 5, 6, 7, 8, 9, ] and references therein. In this paper we are concerned with the following class of reaction-diffusion models: { ut = u xx fu, w = w xx gu, w in the limit and. Under certain general conditions on the nonlinearities f and g that will be specified below, the system admits an interface layer solution. Such a solution has the property that u is either u or for some constants u everywhere except near the interface location, where it has a layer of size O connecting the two constant states u ±. A single mesa or a box solution consists of two interfaces, one connecting to u and another connecting u back to. By mirror reflection, a single mesa can be extended to a symmetric K mesa solution, consisting of K mesas or K interfaces see Figure b. The general system has been thoroughly studied by many authors. In particular, the regime = O is well understood. Under certain general conditions on g and f, it is known that a K-mesa

2 pattern is stable for all K see for example [], []. On the other hand, in the limit =, the system reduces to the so-called shadow system, u t = u xx fu, w ; gu, w =. 3 Equation 3 also includes many models of phase separation such as Allen-Cahn [4] as a special case. Under the same general conditions on g and f, a single interface of the shadow system is also stable; however a pattern consisting of more than one interface is known to be unstable []. The main question that we address in this paper is how the transition from the stable regime = O to the unstable shadow regime = takes place. In a nutshell, when is not too large, the stability of K mesa pattern as shown in [] is due to the stabilizing effect of the global variable w. However as is increased, the stabilizing effect of w is decreased, and eventually the pattern loses its stability. We find that the onset of instability is caused by the interaction between the interfaces of u. This interaction is exponentially small, but becomes important as becomes exponentially large. By analysing the contributions to stability of both w and u, we compute an explicit sequence of threshold values > > 3 >... such that a K mesa solution on the domain of a fixed size R is stable if < K and is unstable if > K. These thresholds have the order ln K = O, so that K is exponentially large in a b Figure : a Coarsening process in Lengyel-Epstein model 8, starting with random initial conditions. Time evolution of u is shown; note the logarithmic time scale. Initial inhomogenuities are amplified due to Turing instability. Shortly after, localized structures are formed at around t =. This is followed by a long transient pattern consisting of three mesas. The middle mesa eventually disappears around t due to the very slow instability of the three-mesa pattern. The remaining two mesas then move towards symmetrical configuration which appears to be stable. The parameter values are =.6, a =, = 5, τ =., domain size is 8. b The snapshot of u at t = 5,,, by which time the system has reached a steady state consisting of a two-mesa pattern. There are many systems that fall in the class described in this paper. Let us now mention some of them. Cubic model: One of the simplest systems is a cubic model, { ut = u xx u 3 w = w xx u β. 4

3 It is a variation on FitzHugh-Nagumo model used in [5] and in [3]. It is a convenient model for testing our asymptotic results. Model of Belousov-Zhabotinskii reaction in water-in-oil microemulsion: This model is a was introduced in [4], [5]. In [6], the following simplified model was analysed in two dimensions: u t = q u xx f w u u q. 5 = w xx uw Brusselator model. In [7] the authors considered coarsening phenomenon in the Brusselator model; a self-replication phenomenon was studied in []. After a change of variables the Brusselator may written as { ut = u xx u uw u 3. 6 = w xx β u Gierer-Meinhardt model with saturation. This model was introduced in [8], see also [9], [] to model stripe patterns on animal skins. After some rescaling, it is u t = u u xx u w κu. 7 = w xx w u This model was also studied in [], where stripe instability thresholds were computed. Lengyel-Epstein model. This model was introduced in [], see also [3]: u t = u xx a 4uv u τv t = v xx uv. 8 u Other models include: models for co-existence of competing species [4]; vegetation patterns in dry regions [5]; models of chemotaxis [6] and models of phase separation in diblock copolymers [7]. Figure a illustrates the instability phenomenon studied in this paper, as observed numerically in the Lengyel-Epstein model. Starting with random initial conditions, Turing instability leads to a formation of a three-mesa pattern at t. However such pattern is unstable, even though this only becomes apparent much later at t. The resulting two-mesa pattern then drifts towards a symmetric position where it eventually settles. A similar phenomenon for the Belousov-Zhabotinskii model 5 is illustrated in Figure 7b. It shows the time-evolution a -mesa solution to 5 with >, starting with initial conditions that consist of a slightly perturbed two-mesa pattern. After a very long time, one of the mesas absorbs the mass of the other, then moves towards the center of the domain where it remains as a stable pattern. In addition to studying the stability of the mesa patterns, we also study their dynamics. This allows us to describe in detail the mechanism by which the exchange of mass between two mesas can take place, as well as the motion of the interfaces away from the equilibrium. For a pattern consisting of K mesas, we derive a reduced problem, consisting of K OE s that gouvern the asymptotic motion of the K interfaces. The outline of the paper is as follows. In we construct the steady state consisting of K mesas. The construction is summarized in Proposition. The main result is presented in 3 Theorem, where we analyse the asyptotics of the linearized problem for the periodic pattern, and derive the instability thresholds K. In 4 we derive the reduced equations of motion for the interfaces. In 5 we present numerical computations to support our asymptotic results. We conclude with discussion of open problems in 6. 3

4 Preliminaries: construction of the Kmesa steady state We start by constructing the time-independent mesa-type solution to. The mesa or box solution consists of two back-to-back interfaces. Thus we first consider the conditions for existence of a single interface solution and review its construction. A mesa solution can then be constructed from a single interface by reflecting and doubling the domain size. Similarly, a K-mesa pattern is then constructed by making K copies of a single mesa. We summarize the construction as follows. Proposition Consider the time-independent steady state of the PE system satisfying { = u xx fu, w = w xx gu, w 9 with Neumann boundary conditions and in the limit Suppose that the algebraic system admits a solution u,, w, with u. efine and suppose in addition that and. fu, w du = ; fu, w = = f, w g ± := g u ±, w f u u ±, w ; and < Then a single interface solution, on the interval [, L] is given by ux U x l g g g <. 3, w w where U is the heteroclinic connection between u and satisfying U yy fu, w = ; U as y ; U u as y ; fu, w = 4 and l is the location of the interface so that { u, < x < l u, l < x < L. Moreover, l satisfies where and l = l = l l O 5 l = g g g L 6 [gu y, w g ] dy [gu y, w g ] dy g g 7 A single mesa solution on the interval [L, L] is obtained by even reflection of the interface solution on an interval [, L] around x =. A K-mesa solution on the interval of size KL is then obtained making K copies of the single mesa solution on the interval [L, L]. 4

5 For future reference, we also define and define constants C ± to be such that µ ± := f u u ±, w ; 8 U y C e µy, U y u C e µy, y ; y. 9 The construction is straightforward and we review it here. First, consider a single interface located at x = l inside the domain [, L]. We assume that u u for < x < l and u for l < x < L where u ± are constants to be determined. Since we assumed that, we expand w = w w so that to leading order w w is constant. Near the interface we introduce inner variables x = l y; ux U x l, w w. Then U y satisfies the system 4 which is parametrized by w. In order for such a solution to exist, u ± must both be roots of fu, w and U must be a hetercolinic orbit connecting u and. This yields the three algebraic constraints which determine u ± and w. To determine the location l of the interface, we integrate the second equation in, and using Neumann boundary condtions we obtain Changing variables x = l y we estimate gu y, w dy l/ lg L gu, w dx =. Ll/ gu y, w dy; [guy, w g ] dy L lg [guy, w g ] dy Expanding l in as in 5 then yields 6 and 7. Since we must have < l < L, this yields an additional constraint 3. 3 Stability of Kmesa pattern We now state the main result of this paper. Theorem Consider the steady state consisting of K mesas on the interval of size KL, with Neumann boundary conditions, as constructed in Proposition. Suppose that Let g w g u f w f u > for all x, and α := C µ 3 f w du exp µ l ; α := C µ 3 f w du f w du g g >. exp µ L l where the constants C ±, µ ±, l are as defined in Proposition. efine Lg := ; 3 g g α 5

6 and for K, define K := L g g g α g α Lg if α α g g α Lg if α α g g α 4 α α cosπ/kgg 4 g α g α if Oα = Oα 4 Then the K mesa pattern is a stable equilibrium of the time-dependent system if < K, and is unstable if > K. Theorem follows from a detailed study of the linearization about the steady state. We consider small perturbations of the steady state of the form ux, t = ux φxe λt, wx, t = wx ψxe λt where ux, wx denotes the K-mesa equilibrium solution of 9 on the interval of length KL with Neumann boundary conditions, whose leading order asymptotic profile was constructed in Proposition. For small perturbations φ, ψ we get the following eigenvalue problem, { λφ = φ f u u, wφ f w u, wψ = ψ g u u, wφ g w u, wψ. 5 with Neumann boundary conditions. The sign of the real part of the eigenvalue λ determines the stability: the system is said to be unstable if there exists a solution to 5 with Reλ > ; it is stable if Reλ < for all solutions λ to 5. To analyse the stability for Neumann boundary conditions, the first step is to consider periodic boundary conditions. The central analysis is summarized in the following lemma. Lemma 3 Periodic boundary conditions Consider the steady state consisting of K mesas on the interval of size KL, as constructed in Proposition, and consider the linearized problem 5 with periodic boundary conditions φl = φkl L The linearized problem admits K eigenvalues. Of these, K are given asymptotically by λ ± θ a ± b f w du L u x 6 where a = α α g g L cosθ g l b = α α α α cosθ g g g g cosθ [ L cosθ ll l ] [ ] L α α lα L lα cosθ 7 8 with and α = C µ 3 f w du exp µ l ; α = C µ 3 f w du exp µ L l ; 9 θ = πk/k; k =...K. 3 6

7 The other two eigenvalues are λ = for which the corresponding eigenfunction is φ, ψ = u, w and g g λ even σ l σ L l f w du L u x 3 where σ ± are given in 4. In the case θ = π, the formula 6 simplifies to λ π α = g L u f w du ; λ π α g g = g L u f w du g g L u x L u x 3 Proof of Lemma 3. The idea is to make use of Floquet theory. That is, instead of considering 5 with periodic boundary conditions on [L, KL L], we consider 5 on the interval [L, L] with the boundary conditions φl = zφl, φ L = zφ L; ψl = zψl, ψ L = zψ L, 33 We then extend such solution to the interval [L, 3L] by defining φx := zφx L for x [L, 3L] and similar for ψ. This extension assures continuity of φ, ψ and φ, ψ at L. Morever, since u, w are periodic with period L, it is clear that φ, ψ extended in this way satisfies 5 on [L, 3L] and moreover φ3l = z φl. Repeating this process, we obtain solution of 5 on the whole interval [L, KL L] with φkl L = φlz K. Hence, by choosing z = exp πik/k, k =...K, we have obtained a periodic solution to 5 on [L, KL L]. To solve 5 subject to 33, we estimate the eigenfunctions as Note that φ c ± u x ; ψ ψ ±l when x ±l. 34 = u xxx f u u x f w w x. Multiplying 5 by u x and integrating by parts on [L, ] we then obtain λc L u x φ x u x φu xx L L f w ψu x φw x We note that the integral term on the right hand side is dominated by the contribution from x = l. Using the anzatz 34 we then obtain λc L Similarly on the interval [, L] we get λc L u x φ x u x φu xx L ψl c w x l f w du 35 u x φ x u x φu xx L ψl c.w x l f w du 36 Write 35 and 36 as c λκ = c κ φu xx L ψl c w x l κ φu xx L ψl c w x l 37 where κ = L u x f w du ; κ = f w du. 7

8 We now transform 37 into an matrix eigenvalue problem. To do so, we will express the boundary terms as well as ψ ±l in terms of c ±. etermining ψ ±l. We start by estimating l l l g u u x dx l g u u x dx g u du g g ; u u g u du g g where ±l ±l denotes integration over any interval that includes ±l and g ± = g u ±, w. On the other hand, φ is dominated by the contribution from the interfaces. Hence we estimate where δ is the delta function. Therefore we write g u φ c g g δ x l c g g δ x l 38 ψ x g g where ηx; x is a Green s function which satisfies c ηx; l c ηx; l with boundary conditions η σx η = δ x x 39 where η L = zηl, η L = zη L, z = exp πik/k, k =...K 4 { σ x σ, x < l σ, l < x < L ; σ f w ± g w g u. 4 f u u=u ±,w=w As will become apparent, the case z = is special and will be considered later. For now, assume z. Then to leading order, η must satisfiy so that η xx = ; η x ; x = η x ; x ; η x ; x η x ; x =. { η A x LB, x < x A L x B Bx x, x > x. The constants A, B are to be chosen so that the boundary conditions 33 are satisfied: A BL L x = za; B = zb We then obtain B = z z ; A = L L x z z ηl; l = ηl; l = Lz z ; 4 Lz zlz ηl; l = z ; 43 ηl; l = Lz l z z = ηl; l. 8

9 In summary, we obtain ψ l ψ l g g ηl; l ηl; l ηl; l ηl; l c c 44 where ηl; l, ηl; l are given by 4 and 43. Boundary terms. Next we compute the boundary terms. We start by estimating the behaviour of u x and φ near L. Since u L =, we have u A[expµ z expµ z], The constant A is found by matching u to the heteroclinic solution as x : Matching 45 and 46 we then obtain z = x L. 45 U y C exp µ y; ux U x l u x l C exp µ. 46 A = C exp µ L l. Performing a similar analysis at x = and at x = L we get: u ±L = C µ exp µ L l ; u = C µ exp µ l. Next we estimate φl. Near x L we write µ φ = C exp x L C exp µ x L where C and C are to be determined. Away from L, we have φ c u. Matching the decay modes, we then obtain µ C = c C exp µ L l. On the other hand, near x L we write µ φ = C 3 exp x L C 4 exp µ x L ; as before, we get µ C 4 = c C exp µ L l. The constants C and C 3 are determined by using the boundary condtions 4, which yields In summary, we get Performing a similar analysis at x, we obtain C 3 = zc ; C 4 = zc µ φl C exp µ [ L l c ] z c ; µ φl C exp µ L l [zc c ]. φ C µ exp µ l [c c ]. 9

10 We thus obtain φu xx L = µ 3 C 3 exp φu xx L = µ 3 C 3 exp so that [ where α ± are given by. Finally, we estimate µ L l [zc c ] C µ 3 [ ] µ L l κ φ x u x φu xx L κ φ x u x φu xx L ] [ = c z c α α α z α 3 exp C µ 3 3 exp µ l µ l α zα ] [ ] c α α c [c c ] ; [c c ] ; w l g l w l. 48 Substituting 47, 48 and 44 into 37 we obtain where λκ c c = a b b a c c a = α α g g ηl; l g l ; b = α zα g g ηl; l. It follows that λκ = a ± b. Next we compute a = α α g g b = α zα g g b = α z α g g Lz z g l Lz zlz z Lz lz z b = α α α g g α z z α 4 g g z L zll l 4 z z We write and note that z z = cosθ ; Lz z l z = e iθ, φ = πk/k z z = cosθ cosθ. α g g L z z l Combining these computations, we obtain 6, 7, 8, provided that z. Next we consider 33 with z =, which corresponds to periodic boundary conditions on [L, L]. This admits two solutions. One is λ = corresponding the odd eigenfunction φ = u x, ψ = w x. The other eigenfunction is even. This corresponds to imposing the boundary conditions φ = = φ L; ψ = = ψ L. 47 As before, we assume φ u x ; ψ ψ l when x l. 49

11 and obtain As before, we obtain λ L u x φ x u x φu xx L ψl w xl f w du 5 ψ x g g ηx; l where ηx; x satisfies 39 with boundary conditions η = = η L. We then obtain η L σx σ l σ L l. The boundary term is evaluated as previously, but is of smaller order. This yields the formula 3 for the even eigenvalue. We now use Lemma 3 to characterize stability with Neumann boundary conditions as follows. Suppose that φ has Neumann boundary conditions on [, a]. Then we may extend φ by even reflection around the origin; it then becomes periodic on [a, a]. From this principle, it follows that the eigenvalues of a K mesa steady state with Neumann boundary conditions form a subset of the eigenvalues of K spikes with periodic boundary conditions. On the other hand, if φ is an eigenfunction on [a, a] with periodic boundary conditions then so is φx and hence ˆφx = φx φx is an eigenfunction on [, a] with Neumann boundary conditions, provided that ˆφx. Since ˆφ = and φ satisfies a nd order OE, ˆφ iff ˆφ iff φ. Verifying this condition, we obtain the following result. Lemma 4 Neumann boundary conditions Consider the steady state consisting of K mesas on the interval of size KL, with Neumann boundary conditions. The linearized problem admits K eigenvalues. Of these, K are given asymptotically by 6 to of Lemma 3, but with θ = πk/k, k =...K. 5 The additional two eigenvalues correspond to an even and odd eigenfunction with Neumann boundary condtions on [L, L]. They are λ odd = with all the symbols as defined in Lemma 3. α g g λ even = σ l σ L l gl u f w du g g L u x L u x Figure shows the actual numerical computation of the four distinct eigenvalues/eigenfunctions for the cubic model 4 with K =. see 5 for methods used. Note that φ is localized at the interfaces and is nearly constant elsewhere; whereas ψ has a global variation. An excellent agreement between the asymptotic results and numerical computations is observed. Critical thresholds. To obtain instability thresholds, we set λ ± θ = in Lemma 3; we then obtain a b =. Using l = gl g g and after some algebra we obtain: f w du = α α cosθ L g α g α g g L g g g g 54 which implies that λ θ = iff > θ where Lg if α α g θ g α Lg if α α g g α 5 53

12 u w 3 φ ψ λ λ even =.557 asympt =.56 λ odd =.9 asympt =.94 λ π/ =.75 asympt =.85 λ π/ =.936 asympt =.95 Figure : Top row: steady-state with two mesas. The cubic model 4 was used with L =, =.3, = 4, β =.3. Bottom four rows: the four possible eigenfunctions and the corresponding eigenvalues. The numerical computations are described in 5 Asymptotic value is computed using 6. Excellent agreement is observed in all cases less than % error.

13 and more generally, without any assumptions on α and α, L θ = g g g α g α 4 α α cosθ g g 4 gα gα. It is clear that θ is an increasing function of θ; it is also easy to verify that λ ± θ < if α ± is decreased sufficiently, or equivalently, if is sufficiently small: in this case the formula 6 reduces to [ λ ± θ κ g g L ± cosθ ld ] cosθ L g l. 55 On the other hand, when K =, the eigenvalues are λ odd and λ even, given by 5, 53. It is clear that λ even < for all ; on the other hand setting λ odd = yields the threshold 3. This completes the proof of Theorem. 4 ynamics We now derive the equations of motion of quasi-stable fronts. This will allow us to describe the dynamics of the fronts that are not necessary in a symmetric pattern. In addition, this will also enable us to describe in more detail the aftermath of an instability of a symmetric pattern. We assume that the pattern consists of K mesas on the interval of length KL. Each mesa is bounded by two interfaces located at x li and x ri and we assume the ordering L < x l < x r < x l < x r < < x lk < x rk < K L. Moreover to leading order we assume { u, if x x u ri, x ri for some i, K, otherwise and near each interface, u x li y Uy, u x ri y Uy, y = O, i =...K, where U is the heteroclinic orbit given in 4, with Uy u ± as y. We also suppose that x li, x ri are slowly changing with time. In addition we define: x ci := x li x ri, i =...K; x di := x ri x li, i =...K ; x d := L, x dk := K L. The equations of motions are derived from K solvability conditions about each interface. First consider the interface x l. We expand where w is given by and ux, t = u z u, wx, t = w w z = x x l t; u z = U z/. Expanding in terms of we obtain = u zz fu, w ; 56 x l tu = u zz f u u, w u f w u, w w ; 57 = w xx gw, u. 58 3

14 We will see later that x l = O so the above expansion is indeed consistent. We multiply 57 by u and integrate on x L, x c. Upon integrating by parts we obtain: x l t xc L u x dx u z u z u u zz x=xc x=l xc L f w w. The boundary term is evaluated similarly as in Section 3. The end-result is, u z u z u u zz x=xc x=l = C µ exp µ x r x l C µ exp µ L x l. The integral terms are estimated as xc L u x dx du dy du dy; dy xc L f w w w x l f w w, udu. Similar analysis is performed at each of the remaining interfaces. In this way, we obtain the following system: x li t BT li w x li u f w w, udu du dy x ri t BT ri w x ri u, i =...K 59 f w w, udu where BT l = C µ exp µ L x l C µ exp µ x r x l 6 BT li = C µ exp µ xl x ri C µ exp µ x ri x li, i =...K 6 BT ri = C µ exp µ x ri x li C µ exp µ xli x ri, i =...K 6 BT rk = C µ exp µ x rk x lk C µ exp µ K L x rk, 63 The constants w x li and w x ri are obtained by recursively solving for w which satisfies: w { = g, x [x li, x ri ], i =...K otherwise g ; w L = = w K L. To simplify the expression for w, we first define the interdistances x l L, i = m i = x li x ri, i =...K ; p i = x ri x li, i =...K. K L x ri, i = K We obtain the following recursion formulae, { w g x li = m, i = w x ri g m i, i =...K w x ri = w x li g p i, i =...K; { m wl g w x li =, i = w x ri w m x ri m i g i, i =...K wx ri = wx li w x li p i g p i, i =...K. 4

15 Expanding, we obtain w x l = wl g m w x r = wl g m m p g p m w x l = wl g m p m m m g p p m w x r = wl g m m p m m m m p m p... The general pattern is wx li = w L g i g i j= wx ri = w L g i g i j= i j= k=j m jm k i j= i k=j p jp k i j= i j= i k=j p jp k i j= i i k=j p jm k i j= p g p m p p p k=j m jp k i m j j= 64 k=j m jm k i i j= k=j m jp k i m j j= i k=j p jm k. 65 i j= It remains to determine the constant w L; this is done by considering the conservation of mass as follows. Integrating the equation for w in we obtain that for all time t, moreover m j = KL p j so that g mj g pj = ; xri x li = KLg g g. 66 ifferentiating 66 with respect to t and substituting into 59 we then obtain, K i= BT ri BT li [w x ri w x li ] f w w, udu =. 67 Substituting 6-63, and into 67 then determines the constant w L. ynamics of a single mesa. For a single mesa, we define x = x lx r to be the midpoint of the mesa. ue to mass conservation, we have x l = x l, x r = x l; l = p j p j g g g L. 68 Substituting 68 and x = x l x r/ into 59 and after some algebra we then obtain, C µ exp µ d dt x L l x C µ exp µ L l x = g u du f w w, udu 69 dy x g g Note that for the special case where C ± = C ; µ ± = µ the formula further simplifies to d dt x = C µ exp µ µ L l sinh du x g u f w w, udu x. 7 g g dy 5

16 5 Numerical computations In this section we validate our asymptotic results by a direct numerical computations of the full system, as well as the linearized equations 5. Let us first describe the numerical methods used. To perform the numerical simulation of the full system we used the standard software FlexPE [8]. It uses a FEM-based approach and automatic adaptive meshing with variable time stepping. We used a global error tolerance of errtol=. which is more than sufficient to accurately capture the interface dynamics [we also verified that changing the error tolerance did not change the solution]. To determine the solution to the linear problem 5, we have reformulated it as a boundary value problem by adjoining an extra equation dλ dx = as well as an extra boundary condition such as ψl =. We used the asymptotic solution derived in as our initial guess. Maple s dsolve/numeric/bvp routine was then used to solve the resulting boundary value problem this routine is based on a Newtontype method, see for example [9]. Unfortunately, we found that sometimes the Newton s method failed to converge, especially for problems with several interfaces. So we resorted to second method: we discretized the laplacian using the standard finite differences, thus converting 5 to a linear algebra matrix eigenvalue problem. On the other hand, this second approach is much less accurate; especially since the required eigenvalue is very small. Because of this, we used the combination of the two approaches: we used method as initial guess to the boundary value problem solver. This finally converged with sufficient precision. 5. Cubic model We now specialize our results to the cubic model 4, f = u 3 w; g = β u. Let us first consider a symmetric mesa solution on interval [L, L], with its maximum at zero. For such a solution, we find w = ; =, u = ; Uy = tanhy; 7 g = β, g = β ; 7 U y = 4 3 ; f w = ; 73 l = β L; l = ; 74 µ ± = ; C ± = ; α ± = 3 exp ± β L. 75 One of the advantages of using the cubic model as a test case is that due to symmetry, l =. This means that the asymptotic results are expected to be very accurate. We obtain the following expressions for λ odd and λ even : λ even 76 λ odd 3 β L 96 exp L 4 β 77 The even eigenvalue λ even is always stable; the odd eigenvalue λ odd becomes unstable as is increased past the critical threshold is given by = β L L exp 8 β. 78 with λ odd < when < and with λ odd > when >. In terms of, we have λ odd 3 β L 4. 6

17 a b 3 4 c 3 4 d Figure 3: a ynamics of a single mesa for the cubic model 4 with with β =.; =., =, L =. Vertical axis is time, horizontal axis is space. The contour u = is shown. Solid lines are the asymptotic results derived in 4. ots represent the output of the full numerical simulation of 4 using FlexPE. The initial conditions are given by 83 with x =.5. The solution moves to the left and converges to a symmetric one-mesa pattern. b Same as in a, but x =.6. The solution moves to the right until it merges with the boundary. c ynamics of two-mesas. The parameters are K =, L = so that x [, 4]; β =.3, =.3, = 7 Initial interface locations are.8,.5,.3, 3.. The solution converges to the symmetric two-mesa solution. d Same as c except that = 85. The two-mesa solution is unstable; the right mesa absorbs the left, eventually resulting in a stable one-mesa pattern which slowly moves to the center. and the equations of motion for a single mesa become dx dt where x = x lx r is the center of the mesa. Note that dx x dt = 3 β L 4 4 sinh 4x x, 79 x = = λ odd so that the linearization of the equations of motion around the symmetric equilibrium agrees with the full linearization of the original PE; the equilibrium x = undergoes a pitchfork bifurcation and becomes unstable as increases past. For K symmetric mesas on the interval of length R, we have L = R/K 7

18 and the thresholds 4 become: K β R K 8 β L 8 exp R K β, if β < ; exp R K β, if β > ; K 8 Finally, if we take the inverted mesa with u near the boundaries, by changing the variables u u, w w, the model remains the same except β is replaced by β. Thus the stability threholds for the inverted mesa are i = β L 8 β R K K i 8 β R K 8 L exp β exp R K β, if β < ; exp R K β, if β > 8 ; K 8 We now numerically validate our asymptotic results by direct comparison with full numerical simulation of the system 4. Mesa dynamics: single mesa. Choose L =, =. and β =.. From 78 we then get c = Now suppose that =. Then the OE 79 admits three equilibria: x = stable and x ± = ±.56 both unstable. We now solve the full system. Take initial conditions to be x x l ux, = tanh tanh x x l ; wx, =. 83 This corresponds to a mesa solution of length l centered at x. Thus if x.56,.56 then we expect the mesa to move to the center and stabilize there. On the other hand, if x >.56 then the mesa will move to the right until it merges with the right boundary. In Figure 3, we plot the numerical simulations for x =.5 and x =.6. The observed behaviour agrees with the above predictions. ynamics of two-mesa solution. Here we consider a two-mesa solution. We take the domain x [, 4] i.e. L =, K = and take β =.3, =.3. From 74, we get l =.35 so that the symmetric equilibrium location of the interfaces are ±.35 and 3±.35 which yields.65,.35,.65, According to 8, the two-mesa symmetric configuration is stable provided that < 8, and is unstable otherwise. To verify this, we solve the full system with initial interface locations given by.8,.5,.3, 3.. These are relatively close to the symmetric equilibrium. We found that when < 8, such configuration converges to the symmetric two-mesa equilibrium; however it is unstable if > 8 see Figure 3c,d. This is in good agreement with the the theoretical threshold = 8. Next we also compute the four eigenvalues for several values of, and compare to asymptotic results, shown in Figure 4. An excellent agreement is once again observed, including the crossing of zero for λ π/ at = 8. The transitional case of β =. This is the degenerate case for which the formula 8 does not apply. In this case, α = α and the formula 4 reduces to K L 56 cos π expl/; β =, K. 84 4K this formula is also valid when K =, as can be verified by comparing it to 78. Note that this is also qualitatively different from β, in that K actually depends on K when β =. To validate 84 numerically, we set =.7, L =, β =. Formula 84 then yields the asymptotic thresholds = 7.8, =., 3 = 9. Next, we have computed the eigenvalues λ π/k explicitly using the full formulation 5 for K =,, 3 several different and for, L as above; these are shown in Figure 5. An excellent agreement can be observed with the predicted threshold values. For example for = 95, a two-mesa solution on the interval of size 4 is stable but a three-mesa on the interval of size 6 solution is not. 8

19 λ even λ odd λ π/ λ π/ Figure 4: The four eigenvalues of the two-mesa pattern of 4 as a function of. Other parameters are as in Figure 3c,d. Circles represent numerical computations of 5; lines are the asymptotic results given by 6. Excellent agreement is observed, including the crossing of λ π/ at = = λ λ π/ λ π/ λ odd..3 a.5.5 β b Figure 5: Instability thresholds of the K-mesa pattern in the cubic model with K =, and 3. a L =, =.7, β =. Circles show λ as computed by numerically solving the full formulation 5 for different values of and the three different modes, as indicated. Solid curves are the asymptotic approximations for λ as given by 6. The K-mesa pattern is unstable for > K where = 7, =, 3 = 9. b The graph of K versus β with L =, =.7, as given by Theorem note the logarithmic scale. The insert shows the zoom near β =. Boundary-mesa versus interior mesas. Let us now compare the stability properties of interior mesas versus patterns with half-mesas attached to the boundary. The latter are equivalent to an inverted mesa patterns. This situation is shown in the Figure 6. Fix =.5, L =. Moreover β =. < so that the roof of the mesa occupies more space than its floor l =.45 < /. In this case, the instability threshold for a single mesa is 78, 3 and for the inverted mesa it is 8, i = 3. Moreover the instability thresholds for K interior mesas on the interval LK with K > is also 8 K 3. This threshold is also the same for two boundary mesas or K inverted mesas Model of Belousov-Zhabotinskii reaction in water-in-oil microemulsion The cubic model is unusual in the sense that due to the symmetry of the interface, the correction to interface length l of Proposition is zero. To see the more usual case when it is not and a dramatic effect that it can have on the accuracy of asymptotics consider the Belousov-Zhabotinskii model 5: 9

20 = 3.4 = 3.78 a 3 b i = 3.78 i = 3.78 c 3 4 d Figure 6: a Single Interior mesa b Two interior mesas c ouble boundary half-mesas, or an inversted single interior mesa d Two half-mesas at the boundaries and one interior mesa, or an inverted two-mesa pattern. In all four cases, β =. and =.5. The instability threshold for is given above the graph. The case a has the biggest stability range here since l < L/. fu, w = f q u q w u ; gu, w = uw; q. 85 As was done in [6], in the limit q, the condition reduces to f w u du f w u u and we obtain to leading order, ; u 3f ; w 4 f /3 as q. in fact, = Oq. To leading order, the profile U then solves U f 4 f /3U U = for y < ; with U = = U and U y = for y > and U u as y. We then obtain { 3f tanh 3 U /4 f /4 / y, y <., y > and g =, g 4f l = L 4f. Next we compute the correction l to interface position using 7. We have = [gu y, w g ] = 4f 4f 4f tanh 3 /4 f /4 / y sech 3 /4 f /4 / y = 3 /4 / 4f 3/4

21 a b Figure 7: a A stable two-mesa solutions to Belouzov-Zhabotinskii model 5. Parameter values are =, =.,, q =. and f =.6. Circles show the full numerical solution. The solid line shows the asymptotic approximation as computed in Proposition, with the mesa half-length l computed to two orders. The dashed line is the same approximation, except l is set to zero. b Time evolution in the BZ model. Parameter values are the same as in a, except for f =.63. Initial conditions were given in the form of a two-mesa asymptotic solution, but shifted to the left by.5. The two-mesa equilibrium appears to be unstable, though the instability is very slow and the two-mesa solution persists until about t E5. This is in good agreement with the theoretical instability threshold of f.6 see text. so that l = 3 /4 / f /4. Finally, we have U 3f tanh 3 /4 f /4 / y 3f 4 exp3 /4 f /4 / y as y so that C = 4 3f ; µ = 3 /4 f /4 / ; f w = u = 3f ; α = /4 f 3/4 3/ expµ l exp µ l = /4 f 3/4 3/ exp4 exp f.75 On the other hand, µ = O /q µ, so that the critical threshold given by 4 becomes K = C 4f f exp f.75 L L ; C e =.8594, K 86 To verify this formula numerically, we set =, =., q =., L = and K =. Next we solved for several different values of f, with initial conditions given by the two-mesa steady state approximation on the interval [, 3], perturbed by a small shift of size.. We found the two-mesa state was unstable with f =.6 or higher but became stable when we took f =.6 or lower see Figure 7.

22 On the other hand, the threshold value as predicted by 86 with above parameter values and K = is f =.64. Thus we obtain an excellent agreement between the asymptotic theory and direct numerical simulations. In Figure 7a, the approximation with and without l to the steady state is shown. We remark that it was essential to compute the correction l to the mesa width; if we were to set l = the constant C =.8594 in 86 would be replaced by.57, a dramatic difference by a factor of e iscussion We have examined in detail the route to instability of the K-mesa pattern of as the diffusion coefficient is increased. The onset of instability occurs for exponentially large ; it is well known that such solution is unstable for the shadow system case =, see []. We have computed explicit instability thresholds K given by Theorem ; as well as mesa dynamics when is large. The instability thresholds are closely related to the coarsening phenomenon see Figure for example. This was previously analysed for the Brusselator in [7], where the authors have guessed at the formula for K, K > given in Theorem by constructing the so-called asymmetric patterns, and without computing the eigenvalues. The formula for K appears to correspond precisely to the point at which the asymmetric solution bifurcates from the symmetric branch [7]. This is indeed the case when µ l µ L l, i.e. Oα Oα. In particular, this is true when either l or L l is sufficiently small. However the study of asymmetric patterns cannot predict the instability thresholds if µ l = µ Ll: in this case, the formula 4 for K actually depends on K, unlike the former case; whereas the construction of asymmetric patterns is K independent. So in the case µ l = µ L l, the full spectral analysis is unavoidable. There are some similarities between the instability thresholds for mesa patterns computed here, and instability thresholds for for Gierer-Meinhardt system computed in [3], [3]. [In [3], a singular perturbation and matrix algebra approach was used; in [3] an approach using Evans functions and Floquet exponents was used. Here, we use a combination of both: singular perturbations and Floquet exponents.] To be concrete, consider the standard GM system, a t = a xx a a /h; = h xx h a. 87 Unlike the K mesa patterns considered in this paper, the steady state for GM system considered in [3] consists of K spikes, concentated at K symmetrically spaced points. The authors derived a sequence of thresholds = exp//5 88 K = [ ], K K ln 89 such that K spikes on the interval [, ] are stable if < K and unstable if > K. By comparison, letting L = K, and considering only the case µ l < µ L l, the thresholds in Theorem become = O K exp K K = O K exp K µ g ; g g µ g g g if µ l < µ L l, K >. Thus K is exponenetially large in for all K, whereas K is O for K > and exponentially large for K =. We remark that the GM model with saturation 7 exhibits mesa patterns when the saturation is sufficiently large, but exhibits spikes when saturation is small. It is an interesting open question to elucidate the transition mechanism whereby a mesa can become a spike and how the various instability thresholds change from being exponentially large to algebraically large as saturation is decreased.

23 The coarsening phenomenon observed in reaction-diffusion systems is also reminiscent of Oswald ripening in thin fluids see for example [3], [33] and references therein. In two dimensions, another instability occurs for radially symmetric spot solutions, see for example [6], [3], [34]. However the instability computed there is initiated because of the curvature of the spot and the instability thresholds occur when = c = O/, with the spot being stable if > c and unstable if < c. Such instability leads to the deformation of the spot into a peanut-like shape and has no analogy to the one dimensional instabilities studied in this paper. Yet, just like in one dimension, it is expected that an interior two-dimensional spot is unstable for the shadow system =. This suggests that there exists in two dimensions a number c > c such that one spot is stable when c, c and is unstable otherwise. We anticipate that just like in one dimension, c would be exponentially large; the exact computation remains an open problem but see [35] for related computations for a spike in GM model. 7 Acknowledgements R.M. would like to thank the support from The Patrick F. Lett Graduate Students Bursary, alhousie. T.K. would like to thank the support from NSERC iscovery grant, Canada. References [] Y. Nishiura and H. Fujii. Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J.Math.Anal., 8:76 77, 987. [] Y. Nishiura and M. Mimura. Layer oscillations in reaction-diffusion systems. SIAM J.Appl. Math., 49:48 54, 989. [3] C. B. Muratov and V. Osipov. Scenarios of domain pattern formation in a reaction-diffusion system. Phys.Rev.E, 54: , 996. [4] J. Rubinstein, P. Sternberg, and J. B. Keller. Fast reaction, slow diffusion, and curve shortening. SIAM J.Appl.Math, 49:6 33, Feb 989. [5] R. E. Goldstein,. J. Muraki, and. M. Petrich. Interface proliferation and the growth of labyrinths in a reaction-diffusion system. Phys. Rev. E, 534: , 996. [6] G. Nicolis and I. Prigogine. Self-organization in Non-Equilibrium Systems. Wiley Interscience, New York, 977. [7] J.. Murray. Mathematical Biology. Springer-Verlag, Berlin, 989. [8] B. S. Kerner and V. V. Osipov. Autosolitons: a New Approach to Problems of Self-Organization and Turbulence. Kluwer, ordrecht, 994. [9] R. Kapral and K. Showalter, editors. Chemical waves and patterns. Kluwer, ordrecht, 995. [] M. Taki M. Tlidi and T. Kolokolnikov, editors. issipative localized structures in extended systems. Chaos Focus issue, 7, 7. [] T. Kolokolnikov, M. Ward, and J. Wei. Self-replication of mesa patterns in reaction-diffusion models. Physica, Vol, 36:4, 7. [] W. M Ni, P. Polácik, and E. Yanagida. Monotonicity of stable solutions in shadow systems, trans. A.M.S, Vol, 35: ,. [3] C. Muratov and V. V. Osipov. General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems. Phys, Rev. E, 534:3-36,

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