Pinned Fronts in Heterogeneous Media of Jump Type

Transcription

1 Pinned Fronts in Heterogeneous Media o Jump Type Peter van Heijster, Arjen Doelman, Tasso J. Kaper, Yasumasa Nishiura, Kei-Ichi Ueda Division o Applied Mathematics, Brown University, 8 George Street, Providence, RI 9, USA Mathematisch Instituut, Universiteit Leiden, P.O. Box 95, 3 RA Leiden, Netherlands Department o Mathematics & Center or BioDynamics, Boston University, Cummington Street, Boston, MA 5, U.S.A. Laboratory o Nonlinear Studies and Computation Research Institute or Electronic Science, Hokkaido University, Sapporo 6-8, Japan Research Institute or Mathematical Sciences, Kyoto University, Kyoto 66-85, Japan October 6, Abstract In this paper, we analyze the impact o a small heterogeneity o jump type on the most simple localized solutions o a three-component FitzHugh-Nagumo-type system. We show that the heterogeneity can pin a -ront solution, which travels with constant non-zero speed in the homogeneous setting, to a ixed, explicitly determined, distance rom the heterogeneity. Moreover, we establish the stability o this heterogeneous pinned -ront solution. In addition, we analyze the pinning o -pulse, or -ront, solutions. The paper is concluded with simulations in which we consider the dynamics and interactions o N-ront patterns in domains with M heterogeneities o jump type N = 3,, M. key words. three-component reaction-diusion systems, pinned ronts, heterogeneous media, geometric singular perturbation theory, semi-strong ront interactions AMS subject classiication. 35K57, 35K5, 35B36, 35B5, 35Q9

2 Introduction In many classical mathematical models, especially those o reaction-diusion type, the medium in which the process under consideration takes place is implicitly assumed to be homogeneous. This is o course a simpliication; natural media, even in their equilibrium states or background states, generally contain heterogeneities. For instance in the ield o superconductivity, there is quite an extensive literature, dating back to the seventies, on the impact o spatially localized heterogeneities, or impurities, on the dynamics o localized structures the so-called luxons, see []. There is also a special interest in the inluence o spatial heterogeneities on the dynamics o localized structures such as ronts and pulses in reaction-diusion equations, see [,, 6, 8,, 5] and the reerences therein. The pinning phenomenon, in which a traveling solution gets trapped by the heterogeneity while its homogeneous equivalent would have kept on traveling, can be considered as one o its most dramatic eects. While pinning has been studied analytically in the special case o scalar equations see [, ] and the reerences therein, it has been studied much less extensively or systems o reaction-diusion equations [9,, ], see Remark.. Numerically, pinning or systems o reaction-diusion equations has been studied more thoroughly. A model or which extensive numerical simulations are available is a generalized FitzHugh-Nagumo-type FHN system. This model has been proposed to describe, on a phenomenological level, the behavior o gas-discharge systems, see [9, ] and reerences therein. In two space dimensions this system is given by U t = D U U + U κ 3 V κ W + κ x, y, τv t = D V V + U V,. θw t = D W W + U W, where U is typically a cubic nonlinearity and κ x, y models the heterogeneity. System. is also a natural extension o the FHN equations to systems with two inhibitors. Since. is a system, the pinning phenomenon cannot be studied by the methods employed in [], which rely heavily on the scalar nature o the equations under consideration. For instance, scalar systems have a gradient structure, and their solutions can be controlled by sub- and super-solutions. These properties are crucial ingredients in the analysis o []. Moreover, unlike the models used or pinning in superconductivity,. is not close to a completely integrable partial dierential equation PDE, so that the impact o localized, spatial heterogeneities cannot be studied by a perturbation method based on this act []. In this paper, we employ a dynamical systems approach to analyze the impact o localized heterogeneities on localized solutions o.. Note that this approach is similar to that o [] which considers a model or Josephson junctions with jump type heterogeneities. In [5], the inluence o a small smoothened jump type heterogeneity on traveling pulses o. in one space dimension is studied. It is observed that a traveling pulse colliding with the heterogeneity can penetrate, be annihilated, rebound, oscillate or get pinned. More precisely, the pinned solutions are observed when the heterogeneity jumps down, and the pinned solutions are in ront o the heterogeneity, that is, they are located immediately to the let o the heterogeneity. In [8], the inluences o two dierent heterogeneities on traveling pulses are investigated or the same model, that is, the inluences o a symmetric bump type, or -jump, heterogeneity and a periodic heterogeneity are studied. For the bump heterogeneity, also penetration, rebound, annihilation, oscillation and pinning are observed. However, there are now two types o pinned solutions. One is

3 x x x Figure : A traveling ront solution o. that asymptotes to a stationary ront, located at the heterogeneity, and hence is said to be pinned there. The system parameters are chosen as ollows α, β, γ, γ, τ, θ, D, ε = 3,,, 3,,, 5,.. The thick black dashed line indicates the location o the heterogeneity. a pinned solution in ront o a bump, and one is a pinned solution in the bump region. The pinned solution which is observed depends on the bump being up or down. For the periodic heterogeneity, also spatio-temporal chaos is observed. In this paper, we present an analytic understanding o the pinning phenomenon in heterogeneous media. That is, we develop a method by which it is possible to predict or which parameter values pinning can be expected. We will ocus on the most elementary localized solutions, that is, the -ront solutions and -ront or -pulse solutions, and we will also use these insights to study more extended patterns. Motivated by the numerical simulations discussed in the previous paragraph, we analyze the pinning phenomenon in the three-component FHN-system. in a particular scaling U t = U ξξ + U U 3 εαv + βw + γξ, τv t = ε V ξξ + U V, θw t = D ε W ξξ + U W, with < ε ; D > ; τ, θ > ; x, t R R + ; α, β R,. γξ = { γ or ξ <, γ or ξ,.3 with γ, R\{}, see Remark.. Here, all parameters are assumed to be O with respect to ε. This typical scaling o. has been introduced in [5]. See Figure or a pinned -ront solution. We choose this particular heterogeneous model since it is suiciently transparent or the purposes o perorming explicit mathematical analysis, while at the same time it is suiciently complex to support complex localized structures. More speciically, our results and analysis simpliy in the special cases when either α = or β =. These correspond to reductions o the 3-component model to a simpler -component model. In particular, our results Theorem. and.5 show that also 3

4 the -component model exhibits pinned ront and -ront solutions. However, as will be clear later on, the dynamics o the -ront solutions and more general N-ront solutions are much richer in the 3-component model, see also [5, ]. Moreover, to analyze scattering and other more complex phenomena observed in [7], the analysis developed here or the ull 3-component model will be required. Also, as is clear rom the above discussion, this model has been extensively studied via numerical simulations. The speciic choice o the heterogeneity comes, on the one hand, rom the jump type heterogeneity considered in [5]. On the other hand, it is motivated by our intention to keep the analysis manageable. Furthermore, a smoothened step unction can be seen as a basic ingredient or more general heterogeneities such as periodic and random media [5]. By the choices we made, we are able or example to explicitly determine the pinning distance o the localized structures to the heterogeneity and how that depends on the system parameters. In [5,, ], a detailed analysis o the existence [5], stability [] and interaction [] o localized structures or the homogeneous problem. with γξ γ, constant is given. O the main results o these papers, three are needed or this paper: Theorem. [] For ε > small enough, the homogeneous model. with γξ γ, constant possesses a stable traveling -ront solution which travels with speed 3 εγ; it possesses no stationary -ront solutions. Theorem. [] For ε > small enough, the width Γ o a -ront solution to the homogeneous model evolves according to Γt = 3 εαe ε Γ + βe ε D Γ γ,. where Γ := Γ Γ, with Γ i the i-th intersection o the Ucomponent with zero. Theorem.3 [5, ] For ε > small enough, the homogeneous model possesses a -parameter amily o stationary -ront, or -pulse, solutions with width ξ γ hom >, i there is a ξγ hom solving This solution is stable i αe εξγ hom + βe ε D ξγ hom = γ..5 αe εξγ hom + β D e ε D ξ γ hom >. Note that the stationary solutions.5 with Γ = ξ γ hom are the ixed points o.. There are several important dierences between the heterogeneous system. and its homogeneous equivalent γξ γ. First, due to the discontinuity in γξ at ξ =, we cannot expect the solutions to the PDE to be smooth. More precisely, due to the heterogeneity the solutions can only be C in time and C in space. Since we are interested in stationary solutions, we can rewrite. into a six dimensional system o ODEs, see or example.. The solutions to these resulting systems will only be C in space. The second, and most important, dierence is that the heterogeneous model is no longer translation invariant. This loss o translation invariance challenges the methods developed in [5, ] in which the existence and stability o localized homogeneous structures was studied. By this loss, the

5 spatial derivative o the localized stationary structure is no longer an eigenunction o the linearized stability problem with corresponding eigenvalue zero, since this derivative is only C and not C. This has two consequences or the stability analysis. First, we have to determine in which ashion this zero eigenvalue perturbs, since it will yield an instability i it moves into the right hal plane. Second, in [] the translation invariance was used to derive an extra solvability condition rom the existence analysis that was used as a crucial ingredient in the stability analysis. Thereore, we have to ind a new way to determine the heterogeneous equivalent o this solvability condition. A third dierence is that, or the heterogeneous system it is a priori not clear what the trivial background solutions are, see also Remark.. However, their asymptotic behavior can be determined. As ξ ±, the background solutions limit on or U, V, W = u ± γ i,,, with u ± γ i = ± εα + β ± γ i + Oε, i =,,.6 see also [5]. Note that u ±, γ i U, V, W = u γ i,,, with u γ i = εγ i + Oε, i =,,.7 are precisely the solutions to the cubic polynomial u u 3 εαu + βw + γ i, i =,. Since the Oε solution.7 is unstable it will not be considered in this paper. System. has two important symmetries and U, V, W, γ, γ U, V, W, γ, γ.8 ξ, γ, γ ξ, γ, γ..9 The irst symmetry allows us to restrict to solutions which asymptote to u γ,, as ξ. For example, we construct stable pinned -ront solutions which asymptote to u γ,,. By the irst symmetry, we then immediately obtain a stability condition or pinned -ront solutions which asymptote to u + γ,, as ξ. The second symmetry allows us to obtain additional results or the mirrored solution by interchanging the role o γ and γ. For example, rom the existence condition or a -ront solution with its ront pinned at the heterogeneity, we obtain the existence condition or a -ront solution with its back pinned at the heterogeneity by interchanging γ and γ. Thereore, without loss o generality we can restrict the numerical simulations to γ > γ as we did in Section.. This paper is organized as ollows. In the next two sections, we establish the conditions under which the heterogeneity pins a traveling -ront solution, and explicitly determine its distance rom the heterogeneity. We recall rom Theorem. that stationary -ront solutions do not exist or the homogeneous problem γ, see Remark.. The main pinning result is, Theorem. For ε > small enough, there exists a stable pinned -ront solution which asymptotes to U, V, W = u γ,, as ξ and whose ront is pinned near the heterogeneity i and only i γ < < γ. 5

6 Figure : Heuristically, we expect pinning o a -ront solution i γ > > γ. That is, we expect pinning in rame II. Note that only the singular limit o the U component o the ront is depicted. See Theorems. and 3. or a more detailed ormulation o this result. Intuitively, this theorem can be explained rom the results on traveling -ront solutions in the homogeneous system, see Theorem.. These -ront solutions which asymptote to U, V, W = u γ,, as ξ travel in the direction o the sign o γ. So, ronts travel to the right i γ is positive and to the let i γ is negative. Thereore, we expect pinning or the heterogeneous model at the heterogeneity i, and only i, γ is positive and γ is negative. Under these conditions, a -ront solution away rom the heterogeneity always moves toward it. All the other conigurations o γ and γ yield movement o the ront toward ininity. See also Figure. In Section, we determine the existence condition or stationary -ront solutions whose ront is pinned near the heterogeneity, where we recall that a -ront solution consists o a ront positive derivatives concatenated to the right with a back negative derivatives. Again, we are able to explicitly compute pinning distance. The main result is, Theorem.5 For ε > small enough and γ > γ, there exists a pinned -ront solution which asymptotes to U, V, W = u γ,, as ξ and whose ront is located near the heterogeneity i and only i there exists a ξ > solving Moreover, the width o the -ront solution is ξ. αe εξ + βe ε D ξ = γ.. See Theorem. or a more detailed ormulation o this result. Note that condition. coincides with the existence condition o a stationary -ront solution in the homogeneous case with γ = γ, see Theorem.3. Also observe the dierent role o the heterogeneity between the pinning o the -ront solution and the pinning o the -ront solution. The heterogeneity creates a new stationary -ront solution, while it selects a particular stationary -ront solution rom a -parameter amily, see Theorems. and.3. Moreover, Theorem.5 conirms the numerical observations o pinned pulses in ront o a heterogeneity which jumps down [5] note the sign dierence in ront o κ x, y and γξ. In this paper, we have rerained rom explicitly determining the stability o this type o -ront solution, especially since the pinning distance is not O, but O log ε, see 6

7 Theorem.. Thereore, determining the stability is a very technical procedure that in principle can be done with the methods discussed in Section 3 o this paper and in []. In the last section, we present numerical results on N-ront solutions with N = 3,, ront solutions or dierent heterogeneities, and traveling -ront solutions τ large. Remark. Note that. diers substantially rom the model studied in [9,, ]. In these papers, the authors analyze the inluence o a spatial heterogeneity in the diusion coeicients on the solutions to a speciic bistable two-component reaction-diusion system. In this paper, the spatial heterogeneity is encoded in the reaction term and. has three components. Moreover, [9,, ] ocus entirely on ront solutions. In [9] pinning, rebound and penetration phenomena are studied in a system that is assumed to be close to a drit biurcation, i.e., the biurcation at which a traveling ront biurcates rom the standing ront. The analysis is based on a center maniold approach that has been developed to describe the weak interactions o ronts [6]. From the analytical point o view, the main dierences between the present work and [9,, ] is that i, we use geometrical singular perturbation theory to explicitly construct the leading order proile o -ront and -pulse, or -ront, solutions and determine explicit stability conditions in a three-component model, see Theorems., 3., and.; ii, our methods enable us to go beyond the setting o weak interactions and thus to consider N-ront patterns N > that interact in a semi-strong ashion [3, ], see also section 5. Remark. In this paper, we only analyze the interaction o the jump heterogeneity through a ast ield o a localized pattern. That is, we only consider patterns that have one o their ronts located near the heterogeneity. We do not analyze the inluence o the jump heterogeneity through the slow ields. That is, we do not analyze stationary ront solutions which are an Oε /, or more, distance away rom the heterogeneity. The numerical simulations in Section indicate that these types o pinned solutions exist or -ront solutions, see Figures and 5. Moreover, the distance between the pinned ront and the heterogeneity diverges as either γ or γ, see. in theorem.. This is related to the act that the homogeneous problem possesses a -parameter amily o stationary -ront solutions or γ =. Thus, as γ i, the ront cannot be considered to be close to the heterogeneity. Thereore, we imposed that γ i is not equal to zero. This way, we make sure that we exclude this type o weak interaction or -ront solutions. Also observe that we do not construct the background solutions, called deect solutions in [5], which can be seen as heteroclinic connections between u γ,, and u γ,, that remain uniormly Oε close to,, : these solutions also interact weakly with the heterogeneity. These weak interactions through the slow ields are the subject o upcoming research. Remark.3 The analysis in this article can be extended to the case in which stripe solutions o equation. are studied on a bounded strip in the plane, that is, x, y R [, L]. For such solutions, one can carry out a modal decomposition in terms o the vertical wave number and then employ the one dimensional analysis used here, incorporating the vertical wave number as a parameter. We reer to [], where such a decomposition and analysis is perormed or stripe solutions in the two dimensional Gierer-Meinhardt model. The interaction with spatial inhomogeneities can also be studied along these lines i the inhomogeneity also is a jump heterogeneity with a vertical structure, that is, i γξ, η = γξ. A more challenging extension to studying. in two dimensions involves examining spot solutions, which are analogs o -ront solutions in two space dimensions. In this regard, we observe 7

8 that the inluence o a jump heterogeneity with a vertical structure on spot solutions o. in two space dimensions has been investigated in [6]. Based on numerical simulations, the spot can or example be attracted to the heterogeneity, and then transported along it. In addition to these numerical results, there has been some recent analysis o spot solutions in the homogeneous two-dimensional version o.. In particular, in [3], the existence and stability o stationary radially-symmetric spot is examined. It is hoped that the analysis in [3] can be extended to spots in the presence o heterogeneities. Existence o pinned -ront solutions In this section, we construct pinned -ront solutions whose ronts are located near the heterogeneity, that is, near ξ =. Theorem. For each D >,τ, θ >, α, β R, γ, R\{}, and or each ε > small enough, there exists a unique pinned -ront solution Ψ ξ which asymptotes to U, V, W = u γ,,.6 as ξ i and only i sgnγ sgnγ. Moreover, the distance ξ Uξ = rom the ront to the heterogeneity is, to leading order, given by ξ = γ + γ arctanh.. γ γ This theorem establishes the existence part o Theorem.. The stability part will be given by Theorem 3. in the next section. Also, this theorem combined with symmetry.8 immediately yields the existence o pinned -ront solutions which asymptote to U, V, W = u + γ,, as ξ, under the same condition sgnγ sgnγ. There are also a couple o special cases. First, i γ = γ, the heterogeneity is symmetric, and we obtain ξ =. Hence, as expected in this special case, the heterogeneity and the location where U crosses zero coincide. Second, the asymptotics o. yields lim γ ξ =, lim γ ξ =. Thus, or decreasing γ i, the location o the ront goes to the let right, respectively boundary o the ast ield I and will eventually move out o the ast ield I as γ i becomes too small. See also Remarks. and 3.. Proo o Theorem.. The method used in [5] to construct stationary pulse solutions o the homogeneous system may be adapted to analyze stationary ronts o.. First, we deine ξ as the unique value o ξ where the U component crosses zero, that is, Uξ =. Note that or the homogeneous problem, this point was not uniquely determined by the translation invariance property. Next, we introduce the ast and slow ields Is :=, ε [ + ξ, I := ] + ξ ε, + ξ, I + ε ε s := + ξ,. These three domains correspond to the slow, ast, and slow regimes observed in the solution dynamics, and in this proo we will analyze the dynamics separately in these domains. 8

9 Since we look or a pinned stationary solution, we can write the heterogeneous problem. as a singularly perturbed 6-dimensional system o irst order ODEs by introducing p, q, r = u ξ, v ξ /ε, Dw ξ /ε: u ξ = p, p ξ = u + u 3 + εαv + βw + γξ, v ξ = εq, q ξ = εv u, w ξ = ε D r, r ξ = ε D w u. Moreover, we know that the heterogeneity lies in the ast ield I, that is, I, because the heteroclinic pinned ront solutions Γ ξ o. whose existence we establish have ronts that lie O close to the heterogeneity. By.3, we can split this non-autonomous system o ODEs into two autonomous systems, one with γξ = γ on ξ, and the other one with γξ = γ on ξ [,. We label these two systems by., respectively,.. The critical points o these systems are, besides the unstable one around zero, given by u, p, v, q, w, r = u ± γ i,, u ± γ i,, u ± γ i,, or i =,, see.6. Since we construct ront solutions that asymptote to U, V, W = u γ,, as ξ, we are actually only interested in the critical points. u, p, v, q, w, r = u γ,, u γ,, u γ, := P,.3 or., and or. we are only interested in u, p, v, q, w, r = u + γ,, u + γ,, u + γ, := P +.. These points are saddle ixed points o. and., respectively. The pinned -ront solutions Γ ξ that we construct with respect to., will lie in the transverse intersection o W u P and W s P + : Γ ξ W u P W s P +..5 Moreover, these solutions will consist o three segments, a let slow segment on I s, a ast segment on I, and a right slow segment on I + s. Finally, since the ront solution Γ ξ is C smooth see Section, the solutions to. and. should match at the heterogeneity, i.e., we must impose lim ξ u, p, v, q, w, rξ = lim ξ u, p, v, q, w, rξ. We begin with the ast segment. The ast reduced system FRS o. is obtained in the limit ε, { uξ = p, p ξ = u + u 3.6, 9

10 and v, q, w, r = v, q, w, r constants. It is independent o γξ, thereore, this FRS coincides with the FRS o. and.. Moreover, it also coincides with the FRS o the equivalent homogeneous problem. Thereore, the leading order results o the homogeneous case apply here, [5]. The FRS system is a conservative system with Hamiltonian Its heteroclinic solutions are u ξ, p ξ = Hu, p = ± tanh u + p ξ, ± u +..7 sech ξ..8 The heteroclinic u ξ, p ξ is relevant to our analysis, since in the ast ield I, the ast part o Γ ξ, i.e., the u and p components o the heteroclinic ront Γ, will be Oε close to u ξ ξ, p ξ ξ, as was also the case in [5]. Next, we turn our attention to the slow ields. We deine the maniolds M ± by M ± = {u, p, v, q, w, r = ±,, v, q, w, r }, the unions o the saddle points o.6 over all possible v, q, w, r. Hence, or ε =, these maniolds are invariant and normally hyperbolic. Now, we turn to the persistence o these slow maniolds or ε > suiciently small. The unction γξ makes. non-autonomous, and hence one cannot immediately apply Fenichel theory [7,, 3] to.. However, since γξ = γ or ξ < along Γ ξ, we are only interested in the persistence o M in the phase space o. and Fenichel theory may be applied directly to the autonomous system. to yield the existence o a slow, invariant maniold M that is Oε close to M, M := { u, p, v, q, w, r u = εαv + βw + γ + Oε, p = Oε }..9 Similarly, since γξ = γ or ξ > along Γ ξ, we are interested in the persistence o M + in the phase space o. and Fenichel theory may be applied directly to the autonomous system. to yield the existence o a slow, invariant maniold M + that is Oε close to M+ See also [5]. M + := { u, p, v, q, w, r u = εαv + βw + γ + Oε, p = Oε }.. In the slow ields I s ±, the heteroclinic orbit Γ has to be exponentially close to M ±,, and its evolution is determined by the slow v, q, w, r equations [5]. These are given by the slow reduced systems SRS o. and., respectively. That is, near M the low is to leading order governed by { vξξ = ε v +, w ξξ = ε D v +,. and near M + by { v ξξ = ε v, w ξξ = ε D v..

11 These lows are independent o γ,, and thus the same as the homogeneous slow lows or the equivalent homogeneous problem. The ixed points o. and. correspond to saddle-saddle points on M ±, which correspond to the background states.3 and., respectively. The result o the theorem will ollow by the Melnikov approach employed in [5], because this calculation will establish that W u M and W s M + intersect transversally. In particular, we determine the net changes in the Hamiltonian.7 and the slow v, w components over the ast ield I or the ull perturbed system.. We start with the latter. Similar to the homogeneous case [5], the v and w components are constant to leading order during the jump through the ast ield I. This can best be seen rom the act that the heterogeneity in γ has no leading order inluence on the v, w equations. More precisely, by. v = v + εq ξ ξ + Oε, w = w + ε D r ξ ξ + Oε in I,.3 with v, q, w, r constants. Hence, on I, the slow components v and w are constant to leading order. Next, we work with Hamiltonian.7. The act that γξ is not constant does not have an eect on the values o the Hamiltonian on the slow maniolds M ±, H M =, ε αv + βw + γ, + Oε 3.. We now can determine the change o the Hamiltonian over the ast ield I in two ways. First, we know that Γ ξ must be exponentially close to M ±, outside the ast ield [5]. Thus, by.5 we have that HΓ ξ H M or ξ Is and HΓ ξ H M + or ξ I s +. Thereore, H = H M + H M = ε γ γ αv + βw + γ + γ + Oε ε,.5 where we have used that the slow components are to leading order constant in the ast ield I. Second, using the act that the ast component o Γ ξ in I is to leading order given by u ξ ξ, p ξ ξ see.8, the change o the Hamiltonian over the ast ield I is also given by H = I H ξ dξ = ε I p ξ ξ αv + βw + γξdξ = εαv + βw + γ ξ u+ ξ ξdξ + εαv + βw + γ u + ξ ξ ξdξ + Oε.6 ε = εαv + βw + εγ tanh ξ + εγ + tanh ξ + Oε ε, where we have decomposed the integral over I into a part governed by. and a part governed by.. Equating.5 and.6, we ind to leading order that αv + βw + γ tanh ξ + γ + tanh ξ =..7 This equation is precisely the Melnikov condition. It establishes a relationship between ξ, v and w or which the maniolds W u M and W s M + intersect transversely. The pinned -ront

12 Figure 3: In the let rame, we plot the most general slow v-dynamics.8. In the middle rame, we set the constant B = A = to obtain the correct asymptotic behavior. In the right rame, we also matched the solution in the ast ield I, see.9. solutions Γ ξ lie inside this intersection, and o all the solutions inside this intersection, they are the only ones that also satisy.5. In act, by imposing.5, we can determine v and w uniquely. Observe that the v, q, w, r components o Γ ξ must be exponentially close to the unstable maniold o the saddle,,, o., as ξ and exponentially close to the stable maniold o the saddle,,, o., as ξ. Thereore, we return to the SRS on the slow maniolds M ±,, see. and.. The solutions to these equations are vξ = { A e εξ + B e εξ in Is, A e εξ + B e εξ + in I s +, and wξ = { C e ε D ξ + D e ε D ξ in Is, C e ε D ξ + D e ε D ξ + in I s +,.8 see Figure 3. Since Γ ξ must asymptote to P ±, we know that B = A = D = C =. Since v, w, as well as their derivatives, do not change to leading order in the ast ield I, see.3, we can match these solutions in the ast ield I. Thereore, we obtain to leading order, A e εξ = B e εξ +, A e εξ = B e εξ, C e ε D ξ = D e ε D ξ +, C e ε D ξ = D e ε D ξ. Thus, A = e εξ, B = e εξ, C = e ε D ξ, D = e ε D ξ, so that the slow v, w components o Γ ξ are in the slow ields given by { { e vξ = εξ ξ in Is, e ε D e εξ ξ + in I s + and wξ = ξ ξ in Is,, e ε D ξ ξ + in I s +.9. Thereore, v := vξ = = wξ = w which also implies that v = w = to leading order. Plugging this into.7 yields., ξ = γ + γ arctanh. γ γ Since the domain o arctanhx is x,, we immediately conclude that ξ is only deined i γ and γ have dierent signs: γ + γ γ γ, = γ γ + γ γ, = γ γ,.

13 This completes the construction o the pinned stationary -ront solutions and hence also o the proo o the theorem. In the case when / I, the ront will move since H.7, unless γ i =, see Remark.. This concludes the existence proo. Remark. The proos o our main results are not presented in ull analytical and especially geometrical details. The analysis can be made rigorous by methods similar to the ones used in [5, ]. However, we rerain rom going into the technical details, since this analysis will provide no additional insight into the dierences between the heterogeneous and homogeneous problems. For the stability analysis o the above constructed stationary -ront solutions, we need additional inormation on their structure, that is, we need the higher order correction term o the ast u component in the ast ield I. This is generally the case or singular perturbed problems o the type studied here. Note that or the homogeneous problem, this additional inormation on the higher order correction term was obtained rom a solvability condition []. However, by the loss o translation invariance, we cannot use this method. First, we introduce a regular expansion o the ast u component o Γ ξ: uξ = u ξ + εu ξ + Oε. Clearly, in the ast ield I, u ξ = tanh ξ ξ.8, which is independent o the heterogeneity. However, the irst order correction term u ξ in the ast ield I depends on the heterogeneity. Lemma. The irst order correction term u ξ o the u component o Γ ξ in the ast ield I is D 3sech ξ ξ + γ [ cosh ξ ξ + sinh ξ ξ cosh ξ ξ + 3 tanh ξ ξ + 3 u ξ ξ sech ξ ξ ] [, ξ ε + ξ,, ξ = D 3sech ξ ξ γ [ cosh ξ ξ + sinh ξ ξ cosh ξ ξ + 3 tanh ξ ξ + 3 ξ ξ sech ξ ξ ] ], ξ [, ε + ξ, with D 3 D 3 = γ γ 3 cosh ξ 3 γ + γ ξ 8.. Note that the condition. will be determined by the matching condition u = u +, where the and + stand or the let, respectively right, limit. In principle, we could determine, the constants D 3 by also matching the derivatives, that is, by imposing u ξ = u ξ +, see.. However, it is not necessary to determine these constants explicitly or the orthcoming stability analysis. Proo. From., we know that the ast equation is u ξξ + u u 3 = εαv + βw + γξ. 3

14 Since the v and w components are zero to leading order in the ast ield I.9, the ast equation in the ast ield I is { εγ, or ξ <, u ξξ + u u 3 = Oε + εγ, or ξ. Clearly, the Oε terms depend on the heterogeneity, and thereore the irst order correction term u ξ o the u component o Γ ξ will also depend on the heterogeneity. We introduce u,i ξ, i =,, such that u,i ξ solves the equation Lu,i := u,i ξξ + u,i 3u,i u = γ i,. with ξ < or i = and ξ or i =. By continuity these two solutions should match in ξ =, u, = u,, u, ξ = u, ξ.. Moreover, by the way Γ ξ is constructed, u, should match up to the irst order correction term o the u component o M at the let boundary o I. Similarly, u, should match up to the irst order correction term o the u component o M + at the right boundary o I. Thus, using.9,., and.9, we obtain to leading order lim ξ u, ξ = lim ξ ξ αvξ + βwξ + γ = γ,.3 lim ξ = lim αvξ + βwξ + γ = γ. ξ u, ξ ξ

15 Since Lp := Lu ξ := Lsech ξ ξ =.8, we recover the expression or ξ., as ollows: = u, Lp = = u, ε +ξ ε +ξ p p ξξ + p 3p u dξ + ε +ξ u, + u, 3u, u [ dξ + ξξ [ ] u, p + ξ ε +ξ [ + u, = γ p ξ ] ε +ξ ε +ξ p dξ + γ + p ξ u, = γ tanh ε +ξ p [ ] ε +ξ u, p ξ ε +ξ p dξ u, p p u, + ξξ p 3p u dξ u, p ξ ] ε +ξ u, + u, 3u, u dξ. ξξ u, ξ ξ + γ + tanh ξ u, ξ + O ε + O ε. To determine u,i ξ, i =,, we use the variation o constants method. To simpliy the calculations and notation, we irst substitute ξ = ξ ξ and make the Ansatz that u,i ξ = C i ξsech ξ + γi, or i =,..5 Plugging these into. and observing that d dξ = d, we ind d ξ with L = γ i = Cĩ ξ ξ sech ξ C ĩ ξ sech ξ tanh ξ + C i Lsech ξ + γi 3γ i tanh ξ,.6 d d ξ + 3 tanh ξ. Since Lsech ξ =,.6 reduces to C ĩ ξ ξ Cĩ ξ tanh ξ = 6γ i sinh ξ..7 We introduce D i ξ as the derivative o C i ξ, and solve the reduced homogeneous version o.7 or D i Di ξ D i tanh ξ = = D i ξ = D i cosh ξ. Applying the variation o constants method or the second time by setting D i ξ = D i ξ cosh ξ as solution o D ĩ ξ Di tanh ξ = 6γ i sinh ξ, 5

16 we obtain Thus, D i ξ cosh ξ = 6γi sinh ξ = D i ξ = γ i tanh 3 ξ + D i. D i ξ = γ i sinh 3 ξ cosh ξ + D i cosh ξ. Integrating this expression once, we ind the expression or C i ξ C i ξ = D3 i + γ i sinh ξ + D i sinh ξ cosh 3 3 ξ + 8 sinh ξ cosh ξ ξ Finally, using the identity and recalling.5, we obtain u,i ξ = C i ξsech ξ + γi + sinh ξ tanh ξ = cosh ξ + sech ξ, = D3sech i ξ + γ i sinh ξ tanh ξ + 8 Di sinh ξ cosh ξ + 3 tanh ξ + 3 ξ sech ξ + γ i = D3sech i ξ + γ i cosh ξ + sech ξ + D i sinh ξ cosh ξ + 3 tanh ξ + 3 ξ sech ξ = D i 3sech ξ + γ i cosh ξ + Di sinh ξ cosh ξ + 3 tanh ξ + 3 ξ sech ξ., i =,..8 Here, Di = 8 Di and D i 3 = D i 3 + γ i, i =,. To determine the constant D, we apply the irst asymptotic condition o.3 to.8. Neglecting exponentially small terms, we obtain γ = lim ξ = lim ξ = lim ξ = lim ξ u, ξ γ cosh ξ + D sinh ξ cosh ξ 3 8 γ e ξ e ξ + + e ξ + D e ξ e ξ 6 8 γ D + γ 3 D which yields that D = γ. Similarly, by the second asymptotic condition o.3, we get that D = γ. The irst matching condition o. gives D 3 D 3 = γ γ cosh ξ γ + γ [ sinh ξ cosh 3 ξ + 3 sinh ξ cosh ξ + ] 3 ξ..9 6

17 Next, we observe that sinh x cosh 3 x + 3 sinh x cosh x = cosh x + cosh x 3 tanh x Thereore, plugging this into.9 using the explicit expression. or tanh ξ, we obtain D 3 D 3 = γ γ cosh ξ 3 γ + γ ξ 8 + γ + γ 3 cosh ξ cosh ξ tanh ξ = γ γ 3 cosh ξ 3 γ + γ ξ 8,. which is.. This completes the proo. Note that we could, in principle, rewrite the cosh -term in. in terms o γ i cosh ξ = γ γ γ + γ. However, we have chosen not do so. 3 Stability o pinned -ront solutions In this section, we determine the stability o the pinned stationary -ront solutions Ψ ξ constructed in the previous section. Theorem 3. For each D >, τ, θ >, α, β R, sgnγ sgnγ and or each ε > small enough, the spectral stability problem associated to the stability o the pinned ront Ψ ξ as established in Theorem. has at most two eigenvalues and the largest eigenvalue is given by λ = 3ε γ γ γ γ + Oε. 3. The essential spectrum lies in the let hal plane Σ := {ω : Rω < χ}, where > χ > max{, /τ, /θ}. Thereore, the -ront solution is stable i and only i γ < < γ. This theorem establishes the stability part o Theorem.. Observe that the stability o the pinned ront is only determined by the orcing parameters γ, γ and is independent o the other system parameters. Also observe that the operator associated to the stability problem is sectorial, so that this spectral stability result implies nonlinear stability. To prove this sectoriality, we note that heterogeneity gives no real extra diiculty and we can ollow the standard methods employed in [8] with a modiication along the lines sketched in [5] since the limit background states o the ront patterns are only known asymptotically in ε, see also []. 7

18 Proo. With abuse o notation, we introduce uξ, vξ, wξ as small perturbations o the stationary -ront solution Ψ ξ = U, V, W ξ, U, V, W ξ, t = U, V, W ξ + u, v, wξe λt. Next, we plug this into the heterogeneous PDE. and linearize to obtain the linear nonautonomous stability/eigenvalue problem. u ξξ + 3U λ u = εαv + βw, v ξξ = ε + τλv u, w ξξ = ε D + θλw u. Note that the inormation about the heterogeneity is encoded only in the non-autonomous term U. Moreover, to leading order, this non-autonomous term is the same as in the homogeneous case, see.8 and []. Thereore, we can immediately conclude two things. First, the essential spectrum o a pinned heterogeneous -ront solution is to leading order the same as or the homogeneous problem. That is, it lies in the let hal plane Σ deined in Theorem 3., see also []. Second, their are at most two point eigenvalues and the largest one is to leading order. Thereore, we rescale λ = ε λ. The other point eigenvalue must, i it exists, lie asymptotically close to 3, the second eigenvalue associated to the ast reduced operator L, see.. Ater rescaling, the ast u equation is Lu = Oε. This yields that u is to leading order given by Cp ξ := C u ξ = ξ Csech ξ ξ, 3. see.8. Hence, the ast u component is to leading order in the slow ields I ± s. Thereore, the slow v, w equations are given by v ξξ = ε v + Oε 3, w ξξ = ε D w + Oε3. These are again independent o the heterogeneity. Moreover, the slow components do not change in the ast ield I to leading order []. Thereore, matching in the ast ield I and imposing the boundary conditions, we ind that v and w are to leading order zero. This implies that we need to rescale v and w: v = εṽ, w = ε w, see also []. The rescaled stability problem is u ξξ + 3U u = ε λu + ε αṽ + β w, ṽ ξξ = εu + ε ṽ + ε 3 τ λṽ, 3.3 w ξξ = ε D u + ε D w + ε3 D θ λ w. We introduce a regular expansion o the ast u component in the ast ield I, that is, uξ = u ξ + εu ξ + Oε, with u ξ given by 3.. Moreover, recall the regular expansion o the stationary U component: U ξ = U ξ + εu ξ + Oε, with U ξ = tanh ξ ξ.8 8

19 and U ξ given by u ξ in Lemma.. The irst order correction term o the ast equation o 3.3 is given by Lu = λu + 6U U u. We can now apply a solvability condition, see also., where we use that u and u ξ continuous in ξ, = I p ξu ξ λ + 6U ξu ξ dξ + Oε = = I sech ξ ξ λ + 6 tanh ξ ξ U ξ dξ + Oε = = 3 λ + 6 I tanh ξ ξ sech ξ ξ U ξdξ + O ε = λ = 9 I tanh ξ ξ sech ξ ξ U ξdξ + O ε. must be 3. Note that or the homogeneous problem this irst order correction term U is zero [5]. This then yields that λ = to leading order and that we need to rescale once more in the homogeneous case. However, due to the heterogeneity this is no longer the case. Thus, the impact o the heterogeneity is that the stability is already determined at an Oε level. Ater implementing U = u, see Lemma., and substituting ξ := ξ ξ, we need to evaluate the ollowing integrals tanh ξ sech 6 ξd ξ = 6 sech6 ξ, tanh ξ sech ξd ξ = sech ξ, tanh ξ sech ξd ξ = 3 tanh3 ξ, tanh ξ sech ξd ξ = 3 tanh3 ξ 5 tanh5 ξ, ξ tanh ξ sech 6 ξd ξ = ξ 6 sech 6 ξ + 3 tanh5 ξ 9 tanh3 ξ + 6 tanh ξ From 3., with ξ = ξ, we obtain λ 9 = D 6 3sech 6 ξ ξ D 6 3sech 6 ε ξ ε ξ γ sech ξ ξ γ sech ε ξ ε ξ γ tanh 3 ξ ξ 6 + γ tanh 3 ε ξ ε ξ γ tanh 5 ξ ξ 3 + γ tanh 5 ξ ε ε ξ 8 γ ξ sech 6 ξ ξ + 8 γ ξ sech 6 ε ξ + ε ξ 8 γ tanh ξ ξ 8 γ tanh ξ ε = 6 sech6 ξ D 3 D 3 γ γ sech ξ γ γ γ + γ 3 tanh3 ξ 8 tanh5 ξ ξ 8 sech 6 ξ + 8 tanh ξ + O ε. ε ξ By. and by the identity, tanh ξ + sech ξ =, 9

20 we obtain that there is one eigenvalue that is to leading order given by λ 9 = γ γ 8 sech6 ξ sech ξ sech ξ + 3 γ + γ tanh ξ 8 tanh ξ + 3 tanh ξ + 8 = γ γ 8 tanh6 ξ + 3 tanh ξ tanh ξ + 6 γ + γ tanh ξ 8 tanh ξ + 3 tanh ξ + 8. Next, we use that tanh ξ = γ+γ γ γ. to derive λ 6 9 = γ γ γ+γ 8 γ γ + γ+γ 3 γ γ γ+γ γ γ γ+γ 8 γ γ + γ+γ 3 γ γ + 8 = γ+γ 6 γ γ γ γ = 3 γ γ γ γ. γ γ + 6 γ+γ Thereore, λ = 3ε γγ γ γ 3., and since by assumption sgnγ sgnγ, we have shown that λ is negative i and only i γ < < γ. This completes the proo, see Remark.. Remark 3. Assume that we have a stable pinned -ront solution Γ ξ located at the heterogeneity γ < < γ, and we increase γ through. Our analysis indicates that the stationary -ront solution will start to travel to the right, and that its speed will approach 3 εγ to leading order []. I we, on the other hand, decrease γ through, then the stationary -ront solution will start to travel to the let with speed asymptoting to leading order to 3 εγ. Existence o pinned -ront solutions In this section, we analyze the existence o pinned -ront solutions. Since we now have two ronts interacting with the heterogeneity and with each other, their are several pinning scenarios. To show some o these several scenarios, we start this section with some numerical simulations. Aterwards, we analyze pinned -ront solutions which asymptote to U, V, W = u γ,,.6 and whose ronts are located near the heterogeneity. Note that rom this result and the system symmetries.8 and.9, we can immediately derive existence results or pinned -ront solutions which asymptote to U, V, W = u ± γ,γ,, and whose ronts or backs are located near the heterogeneity. From the numerical results it ollows that there are also pinned -ront solutions whose ront or back is not located near the inhomogeneity, see the right rames o Figures and 5. However, this type o pinned solution will not be analyzed in this paper, see Remark... Numerical simulations In this section, we look at the inluence o the heterogeneity on the dynamics o -ront solutions which asymptote to U, V, W = u γ,,. Moreover, in all the simulations that ollow the domain

21 o integration is [, ] and we use Neumann boundary conditions. The remaining system parameters α, β, τ, θ, D, ε are kept ixed at 3,,,, 5,.. By the system symmetry.9, we can assume without loss o generality that γ < γ. So, we only have to look at three possible conigurations or the step unction: a γ < γ <, b γ < < γ, c < γ < γ. From Theorem.3 we know that or these system parameters the homogeneous problem possesses a stationary stable -ront solution or < γ < 5 since α = 3, β =. Representative cases are γ = ± and γ = ±. Moreover, rom Theorem. we see or the homogeneous case the competition between on the one hand α and β and on the other hand γ. When these parameters are all positive, the α and β components increase the distance between a ront and a back labeled with Γ, while the γ component decreases this distance. We will also see this competition in some o the simulations o case b and c. We start with case a γ = and γ =. Since both γ i s are negative, the equivalent homogeneous problems possess no stationary stable -ront solutions, see Theorem.3. A ront and a back repel each other as can be explained rom Theorem.. For the heterogeneous problem we observe the same. The ront and back travel in opposite directions toward ininity independent o the initial positions o the ront and back. Note that this behavior is not shown. Next, we analyze case b γ = and γ =. For this step unction there exists a stable pinned -ront solution and an unstable pinned -back solution, see Theorems. and 3.. This is exactly what we observe numerically, i we take the initial position o the back to the right o the heterogeneity ξ >, the ront gets pinned at the heterogeneity, while the back travels to ininity, see the let rame o Figure. However, since to the let o the heterogeneity ξ <, γξ >, we expect that we can ind pinned -ront solutions in this region, see Theorem.3. This is also observed numerically, i we take the initial position o the back to the let o the heterogeneity ξ <, we obtain a pinned solution which is pinned away rom and to the let o the heterogeneity. The width o the pinned -ront solution is ξ = 6, which agrees to leading order in Oε with the predicted homogeneous widths ξ γ hom.5. More precisely, or γ = the predicted homogeneous width is ξhom = 67. Note that the observed distance o the -ront solution to the heterogeneity is ξ = 6. See the right two rames o Figure. For the last case c γ = and γ =, the equivalent homogeneous problems have or both γ i a stable stationary -ront solution, see Theorem.3. So, a priori, we could expect to ind stationary solutions in both regimes. I we take the initial position o the back to the right o the heterogeneity ξ >, the solution asymptotes to a stationary -ront solution whose ront is located at the heterogeneity. The width o the pinned -ront solution is ξ = 3, which agrees to leading order in Oε with the predicted homogeneous widths ξ γ hom.5. More precisely, or γ = the predicted homogeneous width is ξhom = 38. See the let two rames o Figure 5. However, i we take the initial position o the back to the let o the heterogeneity ξ <, we again obtain a pinned solution, but now the solution is pinned away rom and to the let o the heterogeneity. The width o the pinned -ront solution is ξ = 6, which agrees to leading order in Oε with

22 x Figure : In this igure, we look at the inluence o the heterogeneity on the dynamics o -ront solutions. The system parameters α, β, D, τ, θ, ε are ixed at 3,, 5,,,. and the step unction is γ = and γ = case b. The initial positions o the ront and back are dierent in the let and two right rames. In the let rame, the initial position o the back is taken to the right o the heterogeneity and we observe that this back travels to ininity, while the ront evolves to a stationary pinned -ront solution at the heterogeneity. In the middle rame, the initial position o the back is taken to the let o the heterogeneity and the structure evolves to a stationary -ront solution pinned away rom the heterogeneity. The right rame shows the inal structure o this last simulation. The width o the -ront is ξ = 6 and the distance to the heterogeneity is ξ = 6. Note that the thick black dashed line again indicates the position o the heterogeneity x x Figure 5: In this igure, we look at the inluence o the heterogeneity on the dynamics o -ront solutions. The system parameters α, β, D, τ, θ, ε are ixed at 3,, 5,,,. and the step unction is γ = and γ = case c. The initial positions o the ront and back vary in the rames. In the let two rames, the initial position o the back is taken to the right o the heterogeneity and we observe that the structure evolves to a stationary pinned -ront solution at the heterogeneity. The width o the -ront solution is ξ = 3 the second rame shows the inal structure o the simulation. In the right two rames, the initial position o the back is taken to the let o the heterogeneity and the structure evolves to a stationary -ront solution pinned away rom the heterogeneity. The width o the -ront is ξ = 6 and the distance to the heterogeneity is ξ = 5. Note that the thick black dashed line again indicates the position o the heterogeneity.

23 the predicted homogeneous widths ξ γ hom.5. Moreover, the distance o the -ront solution to the heterogeneity is ξ = 5. See the right two rames o Figure 5. These simulations suggest the existence o an unstable scatter solution [8, 5], see also Remark.. Moreover, they suggest, as is also observed in the numerical simulations in [8, 5], that pinned solutions preer smaller γ. That is, we do not ind any pinned solutions whose back is located at the heterogeneity γξ = γ =, but we do ind pinned solutions whose ront is located at the heterogeneity γξ = γ =. Note that this actually ollows rom the existence condition γ > γ o Theorem.5 or.. Finally, note that the distance to the heterogeneity ξ or case b and c dier. This suggests that also or this type o pinned solutions see Remark. the step unction inluences the distance to the heterogeneity ξ.. Existence analysis In this section, we determine the existence condition or pinned -ront solutions whose ronts are located near the heterogeneity and which asymptotes to U, V, W = u γ,,.6. The ideas behind the analysis are similar to those o Section and o [5]. However, since we will ind that ξ but still inside I, i.e., the distance between the ront and the heterogeneity will be and ε / see., we have to perorm a higher order analysis. Nevertheless, we will not go into all the details o the proo, see Remark.. Theorem. For each D >, τ, θ >, α, β R, γ, R\{}, and or each ε > small enough, there exists a pinned -ront solution Ψ ξ which asymptotes to U, V, W = u γ,,.6 as ξ and whose ront is located near the heterogeneity, i and only i γ > γ and there exists a ξ solving αe εξ + βe ε D ξ = γ.. Moreover, the width o the -ront solution is ξ and the distance o the heterogeneity to the ront ξ is given by ξ = log 8 εγ γ.. Note that the existence condition indeed coincides with the existence condition.5 or the homogeneous problem with γ = γ. Moreover, rom the system symmetry.9 we obtain that a pinned solution with its back near the heterogeneity will orm i γ > γ and an existence condition like. with γ replaced by γ. This coincides with the numerical observation that stationary -ront solutions preer smaller positive γ. See also the let rame o Figure 5. Unlike or the -ront solutions, we rerain rom explicitly studying the stability o the pinned -ront solutions. There are two reasons or this: i one does not have to develop new ideas to perorm this stability analysis; ii since we have to consider higher order eects in the existence analysis, we also need to go at least one order higher in the stability analysis, and this requires quite a calculational eort. Proo o Theorem.. The proo will be a combination o the proo o the heterogeneous -ront solution see Section and the proo o the homogeneous -ront solution [5]. The ront sees the heterogeneity, while the back does not see the heterogeneity. We irst deine ξ as the ξ 3

24 value or which the U component crosses zero or the irst time, that is, Uξ =, and U ξ >. Moreover, deine ξ as the second zero o the U component, that is, Uξ =, and U ξ <. By assumption, ξ < ξ, and the width o the -ront is given by ξ := ξ ξ. We introduce three slow ields and two ast ields ], ε + ξ [ ε + ξ, ε + ξ ε + ξ, ε + ξ Is :=, I := ] I := [ ε + ξ, ε + ξ, Is 5 := ε + ξ,,, I 3 s := with I. Next, we reduce the heterogeneous PDE. to the same 6-dimensional system o irst order ODEs. as in Section, and we split it into two dierent system o ODEs. and. as beore. We are interested in solutions Γ ξ which still asymptote to P.3 as ξ. However, we have a dierent asymptotic state at plus ininity see.6. We now have that u, p, v, q, w, r = u γ,, u γ,, u γ, := P, Γ ξ W u P W s P, and again the solutions o. and. should match up at zero. Also the FRS is the same, and thus has the same Hamiltonian structure and the same homoclinic solutions, see.6.8. But now also u + ξ, p + ξ is relevant, since in the second ast ield I uξ, pξ is asymptotically close to u+ ξ ξ, p + ξ ξ [5]. We now need to distinguish between our, instead o two.9., persisting slow maniolds, M i := { u, p, v, q, w, r u = εαv + βw + γ i + Oε, p = Oε }, M + i := { u, p, v, q, w, r u = εαv + βw + γ i + Oε, p = Oε }, i =,..3 In the slow ield Is, the low o. is exponentially close to M. In I3 s, the low is exponentially close to M + and in I5 s, it is exponentially close to M. Moreover, the low on the slow maniolds M, is to leading order given by., while the leading order low on M+, is given by., see [5]. Next, we employ the Melnikov-type approach. In the irst ast ield I, the approach will be the same as that used in Section, since the heterogeneity lies in this ast ield. By contrast, in the second ast ield I, the approach will be similar to that used in [5], since the heterogeneity does not lie in this ast ield. Moreover, the slow v, w components are, to leading order, still constant during the jumps over both ast ields, see.3 and [5]. This results in the eight constants, v, q, w, r = v, q, w, r in I, v, q, w, r = v, q, w, r in I.. The values o the Hamiltonian.7 on the our slow maniolds M ±, are given by., and we determine the change o the Hamiltonian over the ast ields two times in two dierent ways. We start with the ast ield I containing the heterogeneity. Since HΓ ξ H M or ξ Is and HΓ ξ H M + or ξ Is 3, we obtain that during the jump through this ast ield, the Hamiltonian changes in an Oε ashion, H = H M + H M = ε γ γ αv + βw + γ + γ + Oε ε,.5

25 see.5 and.3. On the other hand, since the ast component o Γ ξ in I is to leading order given by u ξ ξ, p ξ ξ see.8, we also obtain H = I H ξ dξ = εαv + βw + εγ tanh ξ + εγ + tanh ξ + Oε ε,.6 see.6. This yields the irst jump condition. To leading order, we have that = αv + βw + γ tanh ξ + γ + tanh ξ..7 Next, we analyze the second jump, the jump through the ast ield I. Since HΓ ξ H M + or ξ Is 3 and HΓ ξ H M or ξ Is 5, we ind that H = H M H M + = Oε ε. On the other hand, since the ast component o Γ ξ in I is to leading order given by u+ ξ ξ, p + ξ ξ see.8, we get H = I H ξ dξ = εαv + βw + γ + Oε ε. This yields the second jump condition. To leading order, we obtain = αv + βw + γ,.8 Note that both jump conditions.7 and.8 depend on ξ and ξ through the slow constants v,, w,.. To determine these slow constants, we return to the slow equations.,. in the slow ields. Ater integrating the equations, implementing the asymptotic behavior and matching the solutions over the ast ields, we obtain e εξ ξ e εξ ξ, ξ Is, vξ = e εξ ξ e εξ ξ +, ξ Is 3,.9 e εξ ξ e εξ ξ, ξ Is 5, and e ε D ξ ξ e ε D ξ ξ, ξ Is, wξ = e ε D ξ ξ e ε D ξ ξ +, ξ Is 3, e ε D ξ ξ e ε D ξ ξ, ξ Is 5.. Thereore, v = v ξ = v = vξ = e εξ ξ and w = w = e ε D ξ ξ. First, we plug the values o these constants into the second jump condition.8 and recall that ξ = ξ ξ, to obtain the existence condition determining the width o the -ront solution, αe εξ + βe ε D ξ = γ, 5

26 see.. Since v = v and w = w, the irst jump condition.7 reduces to leading order to = γ + γ tanh ξ + γ + tanh ξ = = γ γ tanh ξ. Since γ γ, this equation is ulilled only i tanh ξ = to leading order. This implies that ξ. Note that this does not necessarily imply that / I. To ind out, we need to look at the next order in the ast ield I to determine ξ. First, we introduce a regular expansion o the ast u component in the ast ield I [ εu, uξ = tanh ξ ξ + Oε, ξ ε + ξ,, + ] εu, + Oε, ξ [, ε + ξ.. However, it will not be necessary to determine u, and u, explicitly. By.8 and since v = v and w = w,.5 reduces to H = ε γ γ + Oε ε. We obtain the higher order jump condition, by equating this with.6, ε pξαvξ + βwξ + γξdξ = εγ γ + Oε ε.. I Recall rom.3 that in the ast ield I vξ = v + εq ξ ξ + Oε, wξ = w + ε D r ξ ξ + Oε, with q = + v and r = + w, see.9 and.. 6

27 Thereore, the integral o. can be written as I pξαvξ + βwξ + γξdξ = αv + βw + γ tanh ξ ξ +αv + βw + γ tanh ξ ξ +ε α + v β + D + w ξ ξ tanh ξ ξ dξ ξ I +εαv + βw + γ +εαv + βw + γ = γ γ +ε I tanh u, u, ξdξ ξdξ + Oε ε ξ ξ dξ ξ α + v β + D + w ξ ξ tanh ξ ξ +εγ γ u, ξdξ + Oε ε. However, the second and third integrals are zero to leading order. The second integral is zero since the integrand is an odd unction around the center o I. The third integral vanishes to leading order since ξ and thereore u, = u,, see.. So, the only remaining integral must be equal to the right hand side o.. To leading order, we obtain Thereore, to leading order εγ γ = = tanh = tanh = tanh ξ ξ dξ ξ ξ ξ ξ + + Oε e ξ + e ξ + Oε ξ = log 8 εγ γ, dξ ξ dξ ξ dξ ξ 7

28 see.. Note that this ξ is only deined i γ > γ and also note that I. This completes the proo, see Remark.. Lemma. For the heterogeneous problem, there does not exist a pinned -ront solution or which the heterogeneity in γξ lies in between the ront and the back, that is, I 3 s. Proo. An analysis similar to the previous proo gives the ollowing two jump conditions, see.7 and.8 αe εξ + βe ε D ξ = γ, αe εξ + βe ε D ξ = γ,.3 where, as beore, ξ is the width o the -ront solution. Since γ γ, we obtain that.3 has no solutions. 5 Numerical simulations In this section, we show numerical results or several more complex localized structures such as 3-ront solutions, -ront solutions and traveling -ront solutions. We also show the dynamics o -ront solutions or dierent type o heterogeneities, more precisely, or bump, or -jump, heterogeneities and periodic heterogeneities. Also, where possible, we qualitatively explain the observed dynamics or these complex cases rom the -ront and -ront dynamics with step unction heterogeneity derived in the previous sections. It should be noted that the approach developed here can, in combination with the methods o [5,, ], in principle be used to analytically study many o the orthcoming interactions between N-ront patterns and M-jump heterogeneities. However, the calculational complications will rapidly become overwhelming. Nevertheless, our methods in principle enable us to or instance explicitly study the pinning o a 3-ront solution by a -jump heterogeneity see Figure 6c, or that o a - ront by a -jump heterogeneity see Figure 8b. However, it should be remarked that several o the simulations to be presented in this section exhibit a pinning o ronts that takes place in a slow ield. This situation has not yet been studied, see Remark.. Ater also this type o weak interactions has been understood or stationary patterns, one can again in principle take the next step and deduce explicit evolution equations or the interactions between ronts and heterogeneities as was done in [] or the homogeneous problem. Finally, it should be explicitly remarked that the case τ = Oε is still largely ununderstood, even in the homogeneous case, see [5,, ] and Figure. In our simulations, we ocus on the inluence o the heterogeneity. Thereore, we keep most o the other system parameters ixed in this section, i.e., α, β, D, τ, θ, ε = 3,, 5,,,., except in Section 5.5, where we also increase τ to 86. When we use the step unction heterogeneity.3, we can, by the system symmetry.9, restrict ourselves to a ew combinations or γ and γ. Moreover, since the homogeneous problem possesses no stationary solutions i γ > α + β see Theorem.3, we only need to look at a ew dierent cases o sign combinations and magnitudes o γ and γ. Representative cases are given by γ = ± and γ = ±. Typically, the observed dynamics seems to be generic, that is, or slightly dierent parameter values the dynamics is not drastically dierent. 8

29 U.5 x x 8 U - -5 x t U - -5 x t Figure 6: In this igure, we look at the inluence o the heterogeneity on the dynamics o 3-ront solutions. The system parameters α, β, D, τ, θ, ε are ixed at 3,, 5,,,.. The step unction, as well as the initial jump positions vary rom rame to rame. In the let two rames, γ = and γ =, while in the right rame, γ = and γ =. Note that we only plot the U component o the solutions and that thick black dashed line again indicates the position o the heterogeneity. 5. Dynamics o 3-ront solutions We start by analyzing the inluence o the step unction heterogeneity.3 on the dynamics o 3-ront solutions. First, we take a completely positive step unction: γ =, γ =. For the homogeneous problem with γξ = γ or γξ = γ, the utmost right ront travels to plus ininity, and the remaining ront and the back orm a stationary -ront []. This is also what we observe or the heterogeneous case. However, the location o the resulting stationary pinned -ront solution depends on the initial positions o the ronts and back. I the initial position o the right ront is to the right o the heterogeneity, the stationary -ront will be pinned with its back to the heterogeneity, and it has an approximate width o 3, irrespective o the initial location o the let ront and the back, see the two let rames o Figure 6. Note that the resulting stationary pinned -ront solution is as described by Theorem.3 with leading order in ε width 38. For an initial condition with both ronts and the back to the let o the heterogeneity, the stationary -ront will not be pinned at the heterogeneity, but it has the same width 3. This simulation is not shown. So, in all cases, the pinned solution preers the smaller γ i. In the right rame o Figure 6, we changed γ γ, that is, γ = and γ =. We observe that the utmost right ront gets pinned at the heterogeneity, as could be expected rom the existence and stability conditions or pinned -ronts, see Theorems. and 3.. The other ront and the back orm a stationary -ront solution with width 5. Thereore, the heterogeneity is also able to pin 3-ront solutions. However, it also has inluence on the width o the -ront solution. Note that the distance o the back to the pinned ront is 36, so that we actually do have a pinned solution with width as expected rom the existence conditions. 9

30 .5 x 5 3 x 5 8 x Figure 7: In this igure, we look at the inluence o the heterogeneity on the dynamics o the -ront solution. The system parameters α, β, D, τ, θ, ε are ixed at 3,, 5,,,.. The step unction, as well as the initial jump positions vary rom rame to rame. In the let two rames, γ = and γ =. In the right rame, γ = and γ =. Note that we only plot the U component o the solutions and that thick black dashed line again indicates the position o the heterogeneity. 5. Dynamics o -ront solutions Next, we look at the dynamics o -ront solutions. The system parameters are ixed in such a way that there does not exist a stationary -ront solution -pulse or the homogeneous problem independent o the value o γ. See [5], where we prove that α and β need to have dierent signs or stationary -ront solutions. We start with a completely positive step unction, that is, γ = and γ =. We observe that in the simulations the time asymptotic state is such that either all ronts two ronts and two backs are on one side o the heterogeneity see the let rame o Figure 7, or two ronts one ront and one back are to the let o the heterogeneity and two ronts are to the right see middle rame o Figure 7. The dierence in these two runs lies in the initial condition. In the irst case, a stationary and thus pinned -ront solution -pulse solution is ormed although this may be caused by the proximity o the boundary o the domain o integration. The widths o the two pulses are and, respectively. Note that these values dier signiicantly rom the width we expect or a -ront solution, that is, 38. However, the width between the stationary -ronts is again 36. In the second case, the ronts are ostensibly slowly moving in opposite directions and have widths 3 and 5, respectively. These values are both close to the theoretically predicted values or stationary -ront solutions. In the right rame o Figure 7, we took γ = and γ =. The let most ront travels as expected toward, and the let most back gets pinned at the heterogeneity. The remaining ront and back orm a stationary -ront with width 9, as was also observed or the 3-ront case, see the right rame o Figure 6. For a completely negative step unction: γ = and γ =, we observe that the outer ront and back diverge, while the inner ront and back orm a stationary -ront solution mirrored to the ones 3

31 x Figure 8: In this igure, we look at the bump heterogeneity 5. with A = B = 5, the dashed black lines. In the let rame, γ =, γ =, while, in the right rame, γ =, γ =. The other system parameters are ixed at α, β, D, τ, θ, ε = 3,, 5,,,.. Note that we only plot the U component o the solutions. constructed in the previous section. This stationary solution can be located at the heterogeneity or away rom the heterogeneity. Note that this simulation is not shown in Figure A bump, or -jump, heterogeneity In this section, we modiy the heterogeneity rom one step unction to two step unctions, which results in a so-called bump, or -jump, heterogeneity. We look at the inluence on the -ront dynamics, see also [8]. More speciically, { γ, or ξ / [A, B], γξ = γ, or ξ [A, B]. 5. We ocus on the case where sgnγ sgnγ, since or this combination ronts are expected to be pinned, see Theorems. and 3.. More speciically, we set γ = ± and γ =. Moreover, we take A = B = 5, such that a stationary -ront solutions can easily it in the bump region or γξ = γ = ±. First, we set γ = and γ = and take the initial positions o the ront and back in the bump region. The ront and back evolve to a stationary -ront solution with width 5 theoretically 67 and away rom the heterogeneities, see the let rame o Figure 8. So, it appears that the ront and back do not actually see the heterogeneities. This could be expected, since rom Theorems. and 3., we expect that both heterogeneities push the nearby ront or back away. I only one o the initial ronts is in between the heterogeneities, it moves to one o the heterogeneities, while the other ront or back moves toward plus/minus ininity. This run is not shown in Figure 8. For γ = and γ =, the dynamics o the -ront is, as expected, completely opposite. Now, the 3

32 Figure 9: In the let rame, we plotted the evolution o the U component o a -ront solution with a periodic smoothened bump heterogeneity 5.. The dotted lines are the zeroes o γξ. In the middle rame, we plotted the dynamics o the U component o a -ront solution with a sinusoidal periodic heterogeneity γξ = sin.ξ. The other system parameters are ixed α, β, D, τ, θ, ε = 3,, 5,,,.. In the right rame, we plotted this U component ater the timesteps. The dotted line represents the periodic orcing parameter γξ. heterogeneities attract their nearby structure. I the initial jump positions are chosen in between the heterogeneities, a stable stationary -ront solution is ormed at the heterogeneities. The ront is located near the let heterogeneity A, while the back is located at the right heterogeneity B, see the right rame o Figure 8. I only one o the initial ronts is in between the heterogeneities, this ront or back moves to one o the heterogeneities, however, in the opposite direction compared to the previous case γ = and γ =. The ront or back that lies outside the heterogeneities also becomes stationary. This latter simulation is not shown in Figure Periodic heterogeneities In this section, we look at the dynamics o a -ront solution in the presence o a periodic heterogeneity. First, we look at a periodic smoothened bump heterogeneity with period and which oscillates between and, that is, γξ = + i tanh ξ 95 i. 5. i= An initial condition or which the ront and back are relatively close to each other evolves to a stationary solution. However, the ront and the back do not evolve in a symmetric ashion. Moreover, while the back gets pinned at a heterogeneity, the ront is pinned near a heterogeneity, see the let rame o Figure 9. Compare also to the right rame o Figure 8, where the same behavior is observed or the ront and the back. Second, as in [8], we take a sinusoidal periodic heterogeneity. More speciically, the orcing parameter reads γξ = sin.ξ such that γξ [, ] and the period o the unction is π. The ront 3

33 7 x x 6 5 x Figure : In this igure, we plotted the dynamics o a -ront solution with step unction heterogeneity and τ = 86 large. The other system parameters are ixed during the simulation. That is, α, β, D, θ, ε = 3,, 5,,.. Moreover, also the heterogeneity is kept ixed: γ = and γ =. So, the only thing changing rom rame to rame is the initial positions. Note that we only plot the U component o the solutions and that thick black dashed line again indicates the position o the heterogeneity. and the back again move asymmetrically to a stationary solution with ixed width, see the middle rame o Figure 9. In the right rame, we plotted the U component o the inal stationary solution ater time steps together with a part o the periodic orcing parameter γξ dotted line. We observe that or the stationary solution the value o γ at the position o the ront and back are the same and around.85. However, the width o the stationary solution does not coincide with the width o a stationary -ront solution in the homogeneous case with this paricular value o γ. 5.5 Large τ In this last section, we discuss maybe the most interesting case, τ = Oε large, see also []. From the homogeneous analysis [5, ], we know that in this case the stationary -ront solution can biurcate into a uniormly traveling -ront solution or into a symmetrically breathing -ront solution. We consider only one heterogeneity and we keep the system parameters α, β, D, τ, θ, ε positive and ixed at 3,, 5, 86,,.. The most interesting dynamics is observed when both γ i s are positive, that is, γ = and γ =, since in this case the homogeneous equivalent would yield both traveling and breathing -ront solutions. The observed behavior or the heterogeneous case relies heavily on the initial conditions. However, we observe simultaneous traveling and breathing behavior, see the let two rames o Figure. Note that to the right o the heterogeneity the solution seems to travel and breath, while to the let o the heterogeneity it only seems to travel. Moreover, we observe a traveling solution which sometimes bounces o the heterogeneity and sometimes not, see the right rame o Figure. I only one o the γ i s is positive, one o the ronts gets pinned at the heterogeneity, while the other one travels to plus/minus ininity, as could be expected rom the -ront analysis. This be- 33

34 havior is not shown. The homogeneous problem with negative γ possesses no uniormly traveling or breathing -ront solutions [5]. For the heterogeneous problem, we obtain something similar: i both γ i s are negative, the ront and back move in opposite directions toward plus/minus ininity. This simulation is also not shown. Acknowledgments The authors thank P. Zegeling or his valuable assistance with the sotware used in the numerical simulations. PvH and AD grateully acknowledges support rom the Netherlands Organisation or Scientiic Research NWO and the Centrum Wiskunde & Inormatica CWI. PvH also thanks YN and Hokkaido University or their hospitality. TK grateully acknowledges support rom the National Science Foundation through grant DMS YN grateully acknowledges support rom the Grant-in-Aid or Scientiic Research under Grant No.B68. KIU grateully acknowledges support rom the FY 8 Researcher Exchange Program between JSPS and NWO. KIU also thanks AD, PvH and CWI or their hospitality. Reerences [] G. Derks, A. Doelman, S. A. van Gils, and H. Susanto, Stability analysis o π-kinks in a -π Josephson junction, SIAM J. Appl. Dyn. Syst., 6 7, pp. 99. [] N. Dirr and N. K. Yip, Pinning and de-pinning phenomena in ront propagation in heterogeneous media, Interaces Free Bound., 8 6, pp [3] A. Doelman, T. J. Kaper, and K. Promislow, Nonlinear asymptotic stability o the semistrong pulse dynamics in a regularized Gierer-Meinhardt model, SIAM J. Math. Anal., 38 7, pp [] A. Doelman and H. van der Ploeg, Homoclinic stripe patterns, SIAM J. Appl. Dyn. Syst.,, pp. 65. [5] A. Doelman, P. van Heijster, and T. J. Kaper, Pulse dynamics in a three-component system: existence analysis, J. Dynam. Dierential Equations, 9, pp [6] S. I. Ei, H. Ikeda, and T. Kawana, Dynamics o ront solutions in a speciic reactiondiusion system in one dimension, Japan J. Industrial Appl. Mathematics, 5 8, pp [7] N. Fenichel, Persistence and smoothness o invariant maniolds or lows, Indiana Univ. Math. J., 97/97, pp [8] D. Henry, Geometric theory o semilinear parabolic equations, vol. 8 o Lecture Notes in Mathematics, Springer-Verlag, Berlin, 98. [9] H. Ikeda and S. I. Ei, Front dynamics in heterogeneous diuvise media, preprint,. [] H. Ikeda and M. Mimura, Wave-blocking phenomena in bistable reaction-diusion systems, Siam J. On Appl. Mathematics, 9 989, pp

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