Large Stable Pulse Solutions in Reaction-diffusion Equations

Transcription

1 Large Stable Pulse Solutions in Reaction-diusion Equations ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER ABSTRACT. In this paper we study the existence and stability o asymptotically large stationary multi-pulse solutions in a amily o singularly perturbed reaction-diusion equations. This amily includes the generalized Gierer-Meinhardt equation. The existence o N-pulse homoclinic orbits (N 1) is established by the methods o geometric singular perturbation theory. A theory, called the NLEP (=NonLocal Eigenvalue Problem) approach, is developed, by which the stability o these patterns can be studied explicitly. This theory is based on the ideas developed in our earlier work on the Gray-Scott model. It is known that the Evans unction o the linear eigenvalue problem associated to the stability o the pattern can be decomposed into the product o a slow and a ast transmission unction. The NLEP approach determines explicit leading order approximations o these transmission unctions. It is shown that the zero/pole cancellation in the decomposition o the Evans unction, called the NLEP paradox, is a phenomenon that occurs naturally in singularly perturbed eigenvalue problems. It ollows that the zeroes o the Evans unction, and thus the spectrum o the stability problem, can be studied by the slow transmission unction. The key ingredient o the analysis o this expression is a transormation o the associated nonlocal eigenvalue problem into an inhomogeneous hypergeometric dierential equation. By this transormation it is possible to determine both the number and the position o all elements in the discrete spectrum o the linear eigenvalue problem. The method is applied to a special case that corresponds to the classical model proposed by Gierer and Meinhardt. It is shown that the one-pulse pattern can gain (or lose) stability through a Hop biurcation at a certain value µ Hop o the main parameter µ. The NLEP approach not only yields a leading order approximation o µ Hop, but it also shows that there is another biurcation value, µ edge, at which a new (stable) eigenvalue biurcates rom the edge o the essential spectrum. Finally, it is shown that the N-pulse patterns are always unstable when N Indiana University Mathematics Journal c, Vol. 50, No. 1 (2001)

2 444 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER 1. INTRODUCTION In this paper we study the existence and stability o stationary pulse or multi-pulse solutions (U h (x), V h (x)) to a singularly perturbed two-dimensional system o reaction-diusion equations on the unbounded, one-dimensional domain. The unboundedness o the domain relects our choice to study a spatially extended system: the spatial scale o the patterns is much smaller than the length scale o the domain. This implies that the structure and the stability o the patterns does not depend on the domain and/or boundary conditions U,V X FIGURE 1.1. The stable 1-pulse homoclinic solution o the scaled equation (1.7) in the classical Gierer-Meinhardt case (1.9) with µ = 0.38 and ε = 0.1. Note that one has to multiply by 1/ε (1.6) to obtain the amplitudes o the corresponding solution to the unscaled equation (1.2).

3 Large Stable Pulse Solutions in Reaction-diusion Equations 445 The pulse solutions are assumed to be o a homoclinic nature: lim (U h (x), V h (x)) = (U 0,V 0 ), x where (U 0,V 0 ) is an asymptotically stable trivial state that can be set to be (0, 0) by a translation, see Figure 1.1 on the acing page. In the most general setting, the pulse pattern (U h (x), V h (x)) is assumed to be a solution o the system (1.1) { Ut = d U U xx + a 11 U + a 12 V + H 1 (U, V ), V t = d V V xx + a 21 U + a 22 V + H 2 (U, V ), with x R and d V d U, so that we can deine the small parameter ε by ε 2 = d V /d U 1. The nonlinear terms H i (U, V ), i = 1, 2, are assumed to be smooth enough or (U, V ) (0, 0); the coeicients a ij o the linear terms are chosen such that the trivial pattern (U 0,V 0 ) (0,0) is asymptotically stable as a solution to (1.1) with H i (U, V ) 0; see Remarks 1.2 and 1.3. The (generalized) Gierer-Meinhardt system in morphogenesis ([15], [28], and the reerences there) and the Gray-Scott model or autocatalytic reactions ([17], [9], and the reerences there) are among the most well-known examples o systems o the type (1.1) that exhibit (stable) singular pulse patterns o the type studied in this paper. We reer to [27], [24] or many other explicit examples o systems o the type (1.1) originating rom applications in biology, chemistry, and physics. The methods to be developed in this paper can be applied to the existence and stability analysis o singular pulse solutions (U h (x), V h (x)) o (1.1) that can either be large or small (with respect to ε), and positive or negative (see Remark 1.1). In this paper, we will ocus on pulses that are large and positive, i.e., the components o the solutions are positive, U h (x) > 0andV h (x) > 0, and their amplitudes scale with negative powers o ε, see Figure 1.1 on the preceding page. The pulse solutions (U h (x), V h (x)) are also singular in the sense that the V- component evolves on a spatial scale that is shorter than the U-component (since d V d U in (1.1)): V h (x) is exponentially small except on x-intervals that are so short that U h (x) can be assumed to be constant (to leading order) on such an interval, see Figure 1.1. The leading order behavior or U, V 1 o the nonlinear terms H i (U, V ) in (1.1) can be represented by a small number o parameters. Nevertheless, we consider a less general version o (1.1), in order to simpliy the analysis: U t = U xx µu +F 1 (U)G 1 (V ), (1.2) V t = ε 2 V xx V +F 2 (U)G 2 (V ), with µ>0andµ=o(1). The decomposition o the nonlinear terms H i (U, V ) into the products F i (U)G i (V ) is motivated by the act that it is a straightorward procedure to describe the leading order behavior o F i (U)G i (V ) or U and V

4 446 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER large and positive, while this leading order behavior will depend on the relative magnitudes o U and V or general nonlinearities H i (U, V ), see Appendix A on page 501. Thus, the restriction o (1.1) to (1.2) is merely a technical one; it has no inluence on the essence o the method. The same is true or the removal o the linear coupling terms (i.e., a 12 = a 21 = 0) (Appendix A). The critical or signiicant magnitudes o the homoclinic patterns (U h (x), V h (x)) as unctions o ε are determined by a straightorward scaling analysis. We introduce (1.3) Ũ(x,t) = ε r U(x,t), r > 0, Ṽ(x,t) = ε s V(x,t), s > 0, so that the leading order behavior o F i (U) and G i (V ) can be described by the constants α i, β i and i, g i ( 0): (1.4) ( ) ( ) αi Ũ Ũ F i ε r = ε r ( i + ε r F i (Ũ; ε)), ( ) ( ) βi Ṽ Ṽ G i ε s = ε s (g i + ε s G i (Ṽ; ε)), i = 1, 2. Thus, we have implicitly assumed that F i (U) and G i (V ) are smooth unctions or U, V >0 that have a leading order behavior that is algebraic in U, respectively V, when U, V become large. Both in the existence and in the stability analysis o this paper we will have to assume that β i > 1andthatlim V 0 G i (V ) = 0 (i = 1, 2), which implies that the unctions G i (V ) are at least C 1 unctions, or all V. However, the α i s might become negative (in act, we need to impose that α 2 < 0, see Theorem 2.1), so that the unctions F i (U) are allowed to have algebraic singularities as U 0 (see Remark 1.3). The leading order behavior o the nonlinear terms in (1.2) is thus described by r, s, α i, β i, h i, H i,and ˆε, where the latter three expressions are deined by (1.5) ( i + ε r F i (Ũ, ε))(g i + ε s G i (Ũ,ε)) = h i + ˆε H i (Ũ,Ṽ;ε) : h i = i g i, ˆε = ε min(r,s). The details o the scaling procedure are given in Appendix A on page 501. There, it is shown that, with x = ε x, ε = ε,and (1.6) r = β 2 1 D >0, s = α 2 D >0, D = (α 1 1)(β 2 1) α 2 β 1 0,

5 Large Stable Pulse Solutions in Reaction-diusion Equations 447 equation (1.2) can be written in the normal orm 2 U t = U xx ε 2 µu + U α 1 V β 1 (h 1 + ˆεH 1 (U, V; ε)), (1.7) V t = ε 2 V xx V + U α 2 V β 2 (h 2 + ˆεH 2 (U, V; ε)), where we have dropped the tildes. This equation is the subject o the analysis in this paper. An unusual aspect o the pulse solutions under consideration is that they are homoclinic solutions to a point (U, V ) = (0, 0) which is a singularity o the nonlinearities in the equations. While it is possible to control the singularities in the ODE s arising in both the steady state existence analysis and the linearized stability analysis, standard parabolic theory cannot immediately be applied to such basic questions as the well-posedness o the partial dierential equations (1.7) or solutions in a neighborhood o the steady state solutions studied in this paper. The abstract semigroup techniques that are usually employed when deriving nonlinear stability rom linearized stability are similarly unavailable. Standard treatments usually require some amount o smoothness [18]. These are interesting and important issues, but they are not directly related to the present analysis. They are the subject o work in progress, and will be addressed in uture publications; see also Remark 1.3. The singularly perturbed ordinary dierential equation or the stationary solutions to (1.7) can be studied by the methods o geometric singular perturbation theory [11], [21] in combination with the topological approach developed in [8] (or the N>1-pulses). It is shown in Section 5.3 that or any parameter combination in an open, unbounded part o the (α 1,α 2,β 1,β 2,h 1,h 2 )-parameter space (2.5), there exists a homoclinic N-pulse solution (N = 1, 2,..., N =O(1)) or any µ>0 (there is one additional condition on G i (V ), i = 1, 2, or V small, (2.6), see Theorem 2.1). Here, the N describes the number o (ast) circuits the V-component makes through the 4-dimensional phase space; the N-pulse solution is homoclinic to the critical point (0, 0, 0, 0) that corresponds to the trivial state (U 0,V 0 ) (0,0). The unction V h (x) has, as a unction o x, N distinct narrow pulsesatano( log ε ) distance apart. The unction U h (x) is to leading order constant in the region where V h (x) has its peaks, U h (x) decreasesto0onamuch slower spatial scale (Figure 1.1 on page 444). As was the case or a similar analysis in the Gray-Scott model [9], the existence o these solutions depends crucially on the reversibility symmetry (with respect to x) in (1.7) and the act that the limit problem or the slow system is super slow, see Section 2. The geometric analysis also implies that (stationary) N-pulse homoclinic patterns o the type described by Theorem 2.1 can only exist in the non-scaled system (1.2) when U and V are scaled according to (1.3) with r and s as in (1.6). Ater the existence is established, a theory is developed by which the stability o the N = 1-pulse patterns can be determined explicitly as unction o the parameters in the problem (in Sections 3, 4, 5). We will show that the method, called the NLEP approach [6], [7], enables us to determine the number and position (to

6 448 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER leading order) o all elements in the discrete spectrum o the linear eigenvalue problem associated to the stability o the pattern. Here NLEP stands or NonLocal Eigenvalue Problem. The NLEP approach can also be applied to the stability analysis o the N-pulse patterns (N 2), as we shall show in Section 6. The irst step o the NLEP approach (Section 3) ollows the ideas developed by Alexander, Gardner, and Jones ([1], [13]): it is shown that the 4-dimensional linear eigenvalue problem associated to the stability problem or the 1-pulse patterns can be studied through the Evans unction D(λ, ε). D(λ, ε) is in essence the determinant o the 4 4-matrix ormed by 4 independent unctions that span the solution space o the linear eigenvalue problem. By the general theory developed in [1], one can choose these unctions such that D(λ, ε) = 0iandonly i λ is an eigenvalue (counting multiplicities). It ollows rom the singular character o the equations that D(λ, ε) can be decomposed into a product o a ast and a slow transmission component [13]: D(λ, ε) = t 1 (λ, ε)t 2 (λ, ε), where = (λ,µ,ε)>0 is an explicitly known non-zero actor, see (3.21). For N = 1, the ast component, t 1 (λ, ε), corresponds to the Evans unction associated to the stability problem or the stationary homoclinic orbit V red h (ξ) o the scalar ast reduced limit problem (1.8) V t = V ξξ V + h 2 U α 2 0 Vβ 2, with h 2 > 0, β 2 > 1, and U 0 > 0 an explicitly known constant (see (3.13)); ξ = x/ε is the ast spatial scale. Note that this eigenvalue problem can be written as (L (ξ) λ)v = 0, where v(ξ) is deined by V(ξ,t) = V red h (ξ) + eλt v(ξ).it is shown, by a topological winding number argument (as in [1], [13]), that there corresponds to any eigenvalue λ j o this ast reduced stability problem a zero λ j (ε) o t 1 (λ, ε) such that lim ε 0 λ j (ε) = λ j (Section 4). Since it is well-known that the stationary homoclinic solution to (1.8) is unstable with one eigenvalue λ = λ0 > 0 (see also Proposition 5.6), we encounter the same NLEP paradox or this general problem as was studied in [6], [7] or the Gray-Scott model: although t 1 (λ (ε), ε) = 0, we will ind that D(λ (ε), ε) = t 1 (λ (ε), ε)t 2 (λ (ε), ε) 0. The resolution to this paradox lies in a detailed analysis o the slow component t 2 (λ, ε) o D(λ, ε). Using the matched asymptotics approach originally developed in [6], we determine an explicit expression or t 2 (λ, ε) in terms o an integral involving the (uniquely determined) solution to an inhomogeneous version o the reduced ast limit problem: (L (ξ) λ)v = g(ξ), where g(ξ) is an explicitly known unction (see (4.7) and (4.11)). Since the operator (L (ξ) λ) is not invertible at the eigenvalue λ, we deduce that t 2(λ, ε) has apoleoorder1at λ=λ (ε). Hence, D(λ (ε), ε) 0.

7 Large Stable Pulse Solutions in Reaction-diusion Equations 449 Due to the reversibility symmetry x xin (1.2) and (1.7), we know that all eigenunctions v j (ξ) o the ast reduced stability problem are either even (or j even) or odd (or j odd) as unctions o ξ. We show that the zero o t 1 (λ, ε) at λ j (ε) is cancelled by a pole o t 2 (λ, ε) or all even j; however, the inhomogeneous term g(ξ) is such that there is no pole at λ j (ε) when j is odd (Corollary 4.4). This implies that every eigenvalue λ j o the ast reduced limit problem with j odd persists, or ε 0, as an eigenvalue or the ull problem. However, neither o these eigenvalues is positive when N = 1 (see Proposition 5.6), so that we can conclude that all relevant eigenvalues o the ull instability problem are determined by the zeroes o t 2 (λ, ε). Using the leading order approximation o t 2 (λ, ε) we prove, by topological winding number arguments, a number o general results on the instability o the 1-pulse as unction o the parameters µ, α 1, α 2, β 1,andβ 2. The most important o these results, Theorem 5.1, states that there exists a µ U > 0 such that the homoclinic pattern (U h (x), V h (x)) is unstable or all µ<µ U. This or instance implies that the pulse pattern cannot lose its stability by an essential instability, i.e., an instability caused by the essential spectrum (see [37], [38]). Next, it is shown that the inhomogeneous problem ( L (ξ) λ ) v = g(ξ) can be solved explicitly in terms o integrals involving hypergeometric unctions. Such a reduction is known in the literature on (linear) Schrödinger problems in the case that the potential, in our case V red h (ξ), is explicitly known (see or instance [26]). This is only the case in (1.2) or (1.8), when β 2 = 2orβ 2 =3(see[6],which corresponds to β 2 = 2; here V red h (ξ) = c 1/(cosh c 2 ξ) 2 or certain c 1, c 2 ). In this paper, we present a general transormation, that does not depend on an explicit expression or V red h (ξ), by which the inhomogeneous problem can be written as an inhomogeneous hypergeometric dierential equation. This equation can be solved by the classical Green s unction method (see Appendix B on page 502). This solution is substituted into the expression or t 2 (λ, ε). Thus, we have obtained a completely explicit leading order approximation o the slow transmission unction t 2 (λ, ε) that determines the stability o the pattern (U h (x), V h (x)), (5.21). Since we also can determine all zeroes o the ast transmission coeicient t 1 (λ, ε), it ollows that one can obtain all relevant inormation on the spectrum associated to the stability o the 1-pulse patterns by the NLEP approach. It should be noted that the transormation into a hypergeometric orm has also been used to get an exact description o the spectrum o the reduced linear operator L (ξ) λ. It is or instance shown that λ = 1 4 (β 2 + 1) 2 1and that the number o (discrete) eigenvalues equals J + 1, where J = J(β 2 ) satisies J<(β 2 +1)/(β 2 1) J + 1. Hence, J(β 2 ) as β 2 1andJ 1orall β 2 3. See Proposition 5.6. The general theory can be applied to the generalized Gierer-Meinhardt model [20], [28], [29], [30]. This model corresponds to h 1 = h 2 = 1andH 1 (U, V; ε) = H 2 (U, V; ε) 0 in (1.7). Note that the elimination o the higher order nonlinear terms in (1.7) is a rather strong restriction, since the magnitude o these terms can

8 450 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER be O(ε 2 ) (1.5). The special case (1.9) α 1 = 0, α 2 = 1, β 1 = β 2 =2, h 1 = h 2 =1, H 1 (U, V; ε) = H 2 (U, V; ε) 0, in (1.7) corresponds to the original biological values o the parameters by Gierer and Meinhardt [15], see also [20], [28], [29], [30]. It is shown in Section 5.3 that there are two unstable (real) solutions o the equation t 2 (λ, ε) = 0, i.e., eigenvalues, when µ is too small. As µ increases, these solutions merge, at µ = µ complex = (+O(ε)) and become a pair o complex conjugate eigenvalues. This pair crosses the imaginary axis and enters the stable part o the complex plane at µ = µ Hop = (to leading order in ε). Hence, we conclude that the homoclinic 1-pulse solution o the classical Gierer-Meinhardt problem (1.2) with (1.9) is spectrally stable or µ > µ Hop (Theorem 5.11). Moreover, the method also shows that there is a so-called edge biurcation at µ = µ edge = (+O(ε)): a new, ourth, eigenvalue appears rom the edge o the essential spectrum as µ increases through µ edge ; this eigenvalue remains negative or all µ>µ edge. The leading order approximations o µ complex, µ Hop, and µ edge have been computed with the aid o Mathematica; the stability result has been conirmed by a direct numerical simulation o (1.7), see Figure 1.1 on page 444. It is stressed that this result is only an application o the NLEP approach to a special case: the method can be applied to the stability problem or any (singular) homoclinic pattern (U h (x), V h (x)) to (1.2) (and in principle to (1.1), see Appendix A). The stability problem or the N-pulse patterns with N 2 can now be studied along the lines o the machinery developed or the 1-pulse patterns. We can deine the Evans unctions D N (λ, ε) and determine its decomposition into the transmission unctions t1 N (λ, ε) and tn 2 (λ, ε). The most important new insight or this problem is that the linear problem associated to the ast reduced limit problem will have more than one unstable eigenvalue. Intuitively this is clear, since instead o studying the linearization o (1.8) around V red h (x), one now has to linearize around, roughly speaking, N copies o the V red h (x)-pulse (at O( log ε ) distances apart). Thereore, one expects N unstable eigenvalues λ N,j, j = 1, 2,..., N that all merge with λ in the limit ε 0 [35]. It is shown in Section 6 that, or N 2, there is at least one such an eigenvalue, λ N,1, that is associated to an odd eigenunction. By the same mechanism as or the case N = 1, we can conclude that there thus is a λ N,1 (ε) such that t1 N (λn,1 (ε), ε) = 0, while t2 N (λn,1 (ε), ε) is well-deined (i.e., t2 N (λ, ε) has no pole at λn,1 (ε)). Hence, we establish that all N-pulse patterns are unstable or N 2 (Theorem 6.4). Several aspects o the contents o this paper are related to existing literature, such as: the existence and stability o singular localized patterns in the Gray-Scott model [9], [6], [7], [41]; the shadow system approach [20], [28], [29], [30], [2];

9 Large Stable Pulse Solutions in Reaction-diusion Equations 451 the stability o multi-pulse solutions [35], [36], [2], [3]; and the SLEP method [32], [33], [31], [19]. A section in which these relations are discussed concludes the paper. Remark 1.1. We ocus on the existence and stability o large and positive solutions to (1.1) and (1.2) in this paper. However, the methods we develop here can also be used when one is interested in small solutions, or in negative solutions or any mixture (or instance 0 U 1 and V 1). See also Remarks A.1, 2.9, and 3.1. Remark 1.2. I we assume that a ij =O(1), then the solution (U, V ) (0, 0) is asymptotically stable as solution to (1.1), with H i (U, V ) 0 when a 22 < 0, a 11 + a 22 < 0, and a 11 a 22 a 12 a 21 > 0. Remark 1.3. We do allow or singular behavior in the nonlinearities in equations (1.1), (1.2), (1.7) as U 0, as is the case in the (generalized) Gierer- Meinhardt model. We shall see in Section 2 that what makes the singularity at the origin manageable in the existence (ODE) analysis is that the U and V components o the wave decay respectively at slow and ast exponential rates as x. In work in progress, we use semigroup theory in a suitable class o exponentially decaying perturbations to study both well-posedness and nonlinear stability in a neigborhood o the various waves analyzed in the present paper. This technique was irst introduced in [39] or wave solutions o parabolic systems, but a undamental dierence here is that the two components o the wave require dierent exponential decay rates in order to control the singularities in the nonlinear terms. It appears that this leads to a coherent theory o local existence and uniqueness or the partial dierential equations (1.7). The issue o nonlinear stability is more delicate than well-posedness. A new complication arises in the equations o perturbations or the (exponentially) weighted variables, due to the exponential weights that are required to control the singularities at the origin. It turns out that the linearized operator about the wave (in the new, exponentially weighted variables) necessarily has essential spectrum that is tangent to the imaginary axis at the origin, and that this portion o the spectrum cannot be removed by the approach o [39] involving the introduction o additional exponential weights. Hence the general results in [39], [18] still cannot be applied. A similar problem with the essential spectrum occurs in the nonlinear stability analysis o traveling wave solutions o a class o Ginzburg- Landau equations. In [22], a nonlinear stability theory or these solutions or classes o perturbations that decay algebraically at ininity is presented. It appears that the technique developed in [22] is also relevant or the nonlinear stability o the (linearly stable) waves studied in this paper. 2. THE EXISTENCE OF LARGE-AMPLITUDE MULTI-CIRCUIT HOMOCLINIC SOLUTIONS

10 452 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER In this section, we will study the existence o stationary multi-circuit homoclinic solutions to (1.7). The associated ODE can be written in two ways: as the slow system (2.1) u = p p = h 1 u α 1 v β 1 ˆεu α 1 v β 1 H 1 (u, v; ε) + ε 2 µu εv = q εq = v h 2 u α 2 v β 2 ˆεu α 2 v β 2 H 2 (u, v; ε), where denotes the derivative with respect to the slow spatial variable x, andas the ast system (2.2) u = εp ṗ = ε[ h 1 u α 1 v β 1 ˆεu α 1 v β 1 H 1 (u, v; ε)] + ε 3 µu v = q q = v h 2 u α 2 v β 2 ˆεu α 2 v β 2 H 2 (u, v; ε), where denotes the derivative with respect to the ast spatial variable deined by ξ = x/ε. Note that both equations possess the reversibility symmetry (2.3) x, ξ x, ξ, p p, q q. This symmetry will play a crucial role in the orthcoming analysis. The central eature o interest in (2.2) is the semi-ininite, two-dimensional plane (2.4) M={(u,p,v,q):v =q =0,u>0}. Note that the vector ield deined by (2.2) might be singular in the limit u 0(or α i <0, see Remark 1.3). As may be seen rom a direct inspection o (2.2), when ε = 0, this maniold M is invariant under (2.2). In addition, it is also invariant or all ε R, due to the assumptions (1.4) on G i (V ), and the conditions β 1, β 2 > 0 (which we will explicitly assume, see (2.5) in the hypotheses o Theorem 2.1). Finally, there is one ixed point, S, on the boundary o M, precisely at (0, 0, 0, 0). We can now state the main result o this section. Theorem 2.1. Let F i (U) and G i (V ) in (1.2) be such that (1.4) holds with (2.5) 1 g 1 = h 1 > 0, 2 g 2 = h 2 >0, α 1 >1+ α 2β 1 β 2 1, α 2 <0, β 1 >1, β 2 >1,

11 Large Stable Pulse Solutions in Reaction-diusion Equations 453 and let G i (V ) satisy (2.6) G lim i (V ) = 0, i =1,2. V 0 V Then, or any N 1 with N =O(1),(2.2) possesses an N-loop orbit γ N h (ξ) homoclinic to S = (0, 0, 0, 0);theu,vcoordinates o γ N h (ξ) are non-negative; and γn h (ξ) is exponentially close to M, except or N circuits through the ast ield during which γ N h (ξ) remainsatleasto( ε) away rom M. Moreover, each γ N h lies in the transverse intersection o W S (M) and W U (M). This theorem will be proven in Sections The act that the ixed point S lies on the boundary o M, where u = 0, introduces a technical diiculty or the application o the Fenichel geometric singular perturbation theory, since the theory applies or u>0. Moreover, since α 2 < 0, there is a singularity in the vectorield at u = 0 that, a priori, could prevent the existence o orbits homoclinic to S. Both o these issues are treated and resolved in the conclusion o the proo o the theorem in Section 2.4. Remark 2.2. The same geometric method by which the existence o the (multi-pulse) homoclinic solutions is established can also be used to construct several amilies o singular stationary spatially periodic patterns. This has been done in [9], [25] in the case o Gray-Scott model and the ideas developed there can be applied in a straightorward ashion to this more general case. Moreover, unlike the Gray-Scott case, it is also possible to construct singular stationary aperiodic patterns in (2.2). The periodic, respectively the aperiodic, orbits consist o a periodic, resp. arbitrary, arrangement o various kinds o ast N-loop excursions interspersed with long periods close to M. Both types o orbits are exponentially close to certain N-loop homoclinic orbits to M that are not homoclinic to S. Furthermore, it has been shown in [6] that the NLEP approach can also be used, at least on ormal grounds, to study the stability o non-homoclinic solutions (see also [12]). We do not go any urther into this subject o uture research in this paper The geometry o the slow maniolds M and the dynamics o (2.2) on and normal to them. The ast reduced limit ε 0 o (2.2) is: (2.7) v = v h 2 u α 2 v β 2, where u>0 is a constant. System (2.7) is integrable and has a saddle ixed point at (v = 0, v = 0) that has an orbit (v h (ξ), q h (ξ) = v h (ξ)) homoclinic to it when (2.8) and h 2 > 0,

12 454 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER (2.9) β 2 > 1, which are two conditions explicitly contained in the hypotheses (2.5) o Theorem 2.1. Note by (1.6) that the latter condition implies that (2.10) D = (α 1 1)(β 2 1) α 2 β 1 > 0, and that α 2 < 0. We have assumed in the scaling that v =O(1).Wenowseethat even when 0 <v 1, i.e., when 0 <V O(1/ε s ) (and one must analyze the stationary version o problem (1.2) with U = U 0 constant, instead o the reduced system associated to (1.7)), the point (V = 0,dV/dξ =0)is a saddle equilibrium as long as the condition (2.6) ormulated in Theorem 2.1 holds. Also, we note that in this case (when 0 <V O(1/ε s )), there are only two ixed points, i.e., two roots o (2.11) F 2 (U 0 )G 2 (V ) = V, and the homoclinic orbit connecting (V = 0,dV/dξ = 0) is the same as that o (2.7). These statements ollow since U 0 1 (we study large amplitude solutions) implies by (1.4), (1.6), and (2.10) that F 2 (U) 1, and hence, by (2.6), (2.11) can only have solutions or V 1. We can then apply the scaling analysis o Appendix A on page 501 to draw the desired conclusion. The maniold M, deined in (2.4), which is simply the union o the saddle points (0, 0) over all u >0andallp R, is normally hyperbolic relative to (2.2) when ε = 0orallv. Speciically, M has three-dimensional stable and unstable maniolds which are the unions o the two-parameter (u, p) amilies o one-dimensional stable and unstable maniolds, respectively, o the saddle points (v, q) = (0, 0). The Fenichel persistence theory (see [10], [11], and [21]) implies that system (2.2) with 0 <ε 1 has a locally invariant, slow maniold, under the condition that the vector ield is at least C 1. Hence, we have to impose (2.6) (Remark 2.4). Here, we know even more already, since the maniold M is also invariant in the ull system (2.2) with ε 0, as noted above, so that it is a locally invariant, persistent slow maniold. In addition, the Fenichel theory states that, in the system (2.2) with 0 <ε 1, M has three-dimensional local stable and local unstable maniolds, which we denote Wloc(M) S and Wloc(M) U in the ull system, and that these maniolds are O(ε, ˆε) close to their ε = 0 counterparts. The low on M is obtained by setting v, q = 0 in (2.1): (2.12) u = ε 2 µu. Hence, it is linear and slow. Moreover, it is actually super slow, since d/dx =O(ε) and x is already the slow variable. We will ind that this super slowness is crucial both or the existence o the singular pulse solutions and or their stability. Finally,

13 Large Stable Pulse Solutions in Reaction-diusion Equations 455 on M, there are one-dimensional stable and unstable maniolds (restricted to M) that are asymptotic to the saddle S = (0, 0, 0, 0) on the boundary o M: (2.13) l U,S : p =±ε µu. Remark 2.3. The conditions (2.8) and (2.9) arise naturally in the search or homoclinic orbits. First, the condition h 2 > 0 arises since we look or positive solutions u, and so the two terms on the right hand side o (2.7) are o opposite signs, as is necessary. Second, i instead o (2.9) one has 0 <β 2 <1, then M is still invariant but no longer normally hyperbolic, as may be veriied directly on (2.7). Thereore, i the original nonlinearities H 2 (U, V ) o (1.1) or F 2 (U) and G 2 (V ) o (1.2) are such that β 2 < 1 and/or h 2 < 0, then there cannot be a mechanism that enables (positive) solutions to (2.1), (2.2) to be biasymptotic to M. Remark 2.4. Since one has to use the original scalings o (1.2) when v becomes small, it is not necessary to impose the condition β 1 > 1 to apply the persistence results o [10], [11]. Nevertheless, we will see in Section 4.1 that we need to assume that β 1 > 1 in order to develop the NLEP approach (see, however, Remark 3.2). Thereore, we have added this condition to (2.5). Note that the conditions on the G i (V ) s and the β i s in Theorem 2.1 can be replaced by the slightly stronger but simpler assumptions that, or i = 1, 2, G i (V ) = V G i (V ) with G i (0) = 0and G i (V ) algebraically as V. Remark 2.5. When G 2 (V ) V in the limit V 0, i.e., when (2.6) no longer holds, the character o the critical points depends on F 2 (U). We do not consider this case in the paper. Also, the above arguments about G 2 (V ) and the number o ixed points need not hold or general (nonseparable) reaction terms, H 2 (U, V ) F 2 (U)G 2 (V ). In the more general case, one needs to impose another (nontrivial) non-degeneracy condition to avoid additional complications One-circuit orbits homoclinic to M. One-circuit orbits homoclinic to M in the ull system (2.2) with 0 <ε 1will lie in the transverse intersection o the extensions W U (M) and W S (M) o the local maniolds Wloc(M) U and Wloc(M), S and their excursions in the ast ield will lie close to a homoclinic orbit o (2.7) or some particular value o u. In order to detect these solutions, we use a Melnikov method or slowly varying systems. Here, we reine the approach o [34] a little bit so that we have better control over the O( ˆε) terms. In particular, it is helpul to make use o the integrable planar system (2.14) v = v h 2 u α 2 v β 2 ˆεu α 2 v β 2 H 2 (u, v; ε), with q = v,andu>0 ixed, rather than the reduced system (2.7). System (2.14) has a conserved quantity, or energy, given by: (2.15) K(ξ) = 1 2 q2 1 2 v2 + h 2 β uα 2 v β ˆεu α 2 v 0 ṽ β 2 H 2 (u, ṽ; ε) dṽ.

14 456 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER By construction, K M = 0, and K <0 or orbits inside the homoclinic orbit (ṽ h (ξ; u 0 ), q h (ξ; u 0 )) o (2.14), while K>0or orbits outside. In addition, a direct calculation yields: [ (2.16) K = εα2 u α2 1 h2 p β vβ 2+1 v ( + ˆε ṽ β 2 H 2 (u, ṽ; ε) + 1 u H ) ] 2 (u, ṽ; ε) dṽ. α 2 u 0 Hence, K =O(ε), also by construction. Now, since K M 0, any orbit γ(ξ) = (u(ξ), p(ξ), v(ξ), q(ξ)) o (2.2) that is homoclinic to M must satisy the condition (2.17) K(u 0,p 0 )= K(γ(ξ))dξ = 0. Here, without loss o generality, we assume that the orbits γ(ξ) homoclinic to M, i they exist, are parameterized such that γ(0) = (u 0,p 0,v 0,0). Thereore, (2.16) implies: [ v ( (2.18) εα 2 u α2 1 h2 p β vβ ˆε ṽ β 2 H 2 (u, ṽ; ε) u H 2 (u, ṽ; ε) α 2 u ) ] dṽ dξ = 0. The condition (2.18) is exact in the sense that we did not introduce any approximations so ar. Moreover, as we now show, i the zero o K is a simple one, then the homoclinic orbit γ(ξ) lies in the transverse intersection o W S (M) and W U (M). Now, W S (M) and W U (M) are three-dimensional maniolds. Thus, in the our-dimensional phase space o (2.2), one expects that W S (M) W U (M) is a two-dimensional maniold, or, equivalently, that there is a one-parameter amily o orbits γ that are homoclinic to M. The analysis carried out in the remainder o this subsection reveals that this is indeed the case (i (2.5) holds). Since W S (M) and W U (M) are O(ε, ˆε) close to the (u 0,p 0 )-amily o homoclinic orbits to (2.7), as stated above, both W S (M) and W U (M) intersect the three-dimensional hyperplane {q = 0} transversely in two-dimensional maniolds, deined as I 1 (M) and I +1 (M), respectively. These maniolds can be parameterized by (u 0,p 0 ): (2.19) I ±1 (M)={(u 0,p 0,v ±1 0 (u 0,p 0 ), 0), u 0 > 0} {q=0}. Thus, or every u 0 > 0andp 0 R, there exists a v0 1 such that the solution γ(ξ) o (2.2) with γ(0) = (u 0,p 0,v0 1,0)satisies lim ξ γ(ξ) M. Similarly,

15 Large Stable Pulse Solutions in Reaction-diusion Equations 457 there exists a v 0 +1 such that the solution γ(ξ) o (2.2) with γ(0) = (u 0,p 0,v 0 +1,0) satisies lim ξ γ(ξ) M(where the superscripts are indices, not powers). Note that using the above limits is, in act, a slight abuse o notation. The slow maniold M has a boundary M ={(u,p,v,q):u=v =q =0}(2.4) and the vector ield (2.2) can be singular when u 0. However, in this paper we are only interested in orbits that are homoclinic to S M and we will show in Section 2.4 how to extend the geometric analysis to the boundary o M. Having established that the sets I 1 (M) and I +1 (M) are nonempty, we now show they intersect in the hyperplane {p = 0}. We remark that we have a choice in how to show this. We can use either the homoclinic orbit (v h (ξ), q h (ξ)) o (2.7) or the homoclinic orbit (ṽ h (ξ), q h (ξ)) o (2.14) to approximate the solutions on W U (M) and W S (M), and we choose the latter, as is consistent with our choice o K. In particular, using ṽ h to approximate the ast-ield behavior o the solutions to (2.2) in W U (M) and W S (M), the exact condition (2.18) implies that, to leading order, one-circuit homoclinic solutions must satisy: [ ṽh ( (2.20) εα 2 u α 2 1 h2 0 p 0 β ṽβ 2+1 h + ˆε ṽ β 2 H 2 (u 0, ṽ; ε) 0 ) + 1 α 2 u 0 H 2 u (u 0, ṽ; ε) ] dṽ dξ +O(ε 2 ) = 0. The improper integral exists because ṽ h converges exponentially to zero as ξ ±, and hence so does the entire integrand. Then, since the integrand in (2.20) is positive and since u 0 is assumed to be positive, we see that it is only possible to satisy (2.20) i p 0 is O(ε). In addition, we conclude that (2.21) K(u 0,p 0 )=O(ε 2 ) or p 0 =O(ε). Finally, or the one circuit homoclinic orbits we are ater here, we now show that not only is it necessary that p 0 =O(ε), but it is in act the case that p 0 0. We go back to the exact condition (2.18). For any solution γ(ξ) = (u(ξ), p(ξ), v(ξ), q(ξ)) W U (M) with γ(0) = (u 0, 0,v +1 0 (u 0, 0), 0) I +1 (M), the reversibility symmetry (2.3) implies that (2.22) u( ξ) = u(ξ), v( ξ) = v(ξ), p( ξ) = p(ξ), q( ξ) = q(ξ), and, hence also, v 0 +1 = v0 1. Thereore, along γ(ξ), the integrand in (2.18) is an odd unction o ξ, and the integral vanishes identically. This, in turn, implies that W U (M) W S (M) precisely in the orbit γ(ξ) and that the set I +1 (M) I 1 (M) {p = 0}.

16 458 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER 2.3. Multi-circuit orbits homoclinic to M. In this subsection, we extend the results or one-circuit homoclinic orbits to M rom the previous subsection to multi-circuit orbits homoclinic to M. As we shall show below, the global stable and unstable maniolds o M intersect the hyperplane {q = 0} many times. The sets I ±1 (M) deined above can be seen to be the irst intersections o W S (M) and W U (M) with {q = 0}: an orbit γ(ξ) with initial condition in I 1 (M) only ollows the reduced ast low or hal a circuit and gets caught by M, i.e., it does not leave an exponentially small neighborhood o M anymore. Orbits that have their initial conditions in the second intersections o W U,S (M) with {q = 0}, whose existence we shall show shortly, ollow the ast low or two hal circuits through the ast ield beore settling down on M. Hence, they each make one ull circuit. We label these sets o initial conditions (u 0,p 0,v ±2 (u 0,p 0 ), 0) by I ±2 (M). For initial conditions in them, v 0 ±2 (u 0,p 0 )are strictly O( ε). Similar deinitions can be given or the n-th intersection sets I ±n (M). These sets are also two-dimensional maniolds. For n even, v 0 ±n is strictly O( ε), since these solutions make n/2 ull circuits in the ast ield; while, or n odd, v 0 ±n is strictly O(1) (and O(ε, ˆε) close to the intersection o the corresponding unperturbed homoclinic orbit o (2.7) with {q = 0}), since these solutions make a hal-integer number o circuits in the ast ield. Below, we will show that all I ±n (M) exist. Finally, we will show that I m (M) I n (M)or all m + n even, and it is precisely in these intersections in which the orbits homoclinic to M, that make (m + n)/2 ull circuits through the ast ield o (2.2), lie. Remark 2.6. Intersections with m +n odd are ruled out due to the locations o v0 m and v0 n, since one o these is strictly O( ε), while the other is strictly O(1). We irst establish that the curves I ±n (M) exist or all n>1 (i (2.5) holds), ocusing on the case o I +n (M), since the case o I n (M) may be done similarly. The plane {p 0 = 0} separates I +1 (M) into two parts. Orbits with initial conditions in the wrong part o I +1 (M) are outside the three-dimensional maniold W S (M) and ollow the unbounded part o the integrable low (2.14) in orward time ξ. Hence, they do not return to {q = 0}. On the other hand, orbits with initial conditions in the right part o I +1 (M) are inside the three-dimensional maniold W S (M) and ollow the bounded part o the integrable low (2.14) in orward time ξ. Hence, there is the possibility that they can return to {q = 0}. In order to deduce which part o I +1 (M) does return to {q = 0}, i.e., which part o I +1 (M) is the right part, we consider an orbit γ +1 (ξ) = (u +1 (ξ), p +1 (ξ), v +1 (ξ), q +1 (ξ)) with γ +1 (0) = (u +1 0,p+1 0,v+1 0,0) I+1 (M). We assume that p 0 +1 is strictly O(ε), i.e., γ +1 (0) is not too close to I 1 (M). Thus,γ +1 is at its minimal distance

17 Large Stable Pulse Solutions in Reaction-diusion Equations 459 (O( ε))rommwhen ξ = Ξ =O( log ε ). Then, since γ +1 (ξ) Mas ξ and since K 0onM,weseethat (2.23) K(γ +1 (Ξ)) = Ξ K(γ(ξ))dξ = ε α 2h 2 β (u+1 0 )α 2 1 p 0 +1 ṽ β 2+1 h dξ +O(ε 1+σ ) or some σ > 0(sinceṽ h (ξ) approaches 0 exponentially ast), where we have made the same approximation as in (2.20). Thus, since h 2 > 0andα 2 <0 (1.6), (2.10), we have (2.24) K(γ +1 (Ξ)) < 0 p +1 0 > 0. Finally, since K <0, we know rom the deinition (2.15) o K that γ +1 (Ξ) is inside W S (M); and, also that γ +1 (Ξ) intersects {q = 0} again, i.e., I +2 (M) is nonempty. Correspondingly, the above argument shows that i p 0 +1 < 0, then K > 0 and the orbit γ +1 (Ξ) is outside W S (M). Hence, it cannot intersect the hyperplane {q = 0} again. The same argument in backwards time yields that orbits γ 1 (ξ) with γ 1 (0) = (u 1 0,p 1 0,v 1 0,0) I 1 (M) intersect {q = 0} again when p0 1 < 0 (but not when p0 1 > 0). Thus, both I ±2 (M) exist. Furthermore, the above argument may be extended to show that all I ±n (M) exist, and we denote the points in these sets by (u ±n 0, 0,v±n 0 (u ±n 0, 0), 0), respectively. Next, we show that the intersections I +2 (M) I 2 (M), and their higher order equivalents, exist. Orbits with initial conditions in I ±2 (M) can also be approximated by ṽ h (ξ) (2.14) to leading order, since both circuits must be O(ε) close to ṽ h (ξ). Thus, to leading order, (2.25) K = 2ε α 2h 2 β (u+1 0 )α 2 1 p 0 +1 ṽ β 2+1 h dξ, and we ind that the p-coordinate p 0 o the initial condition must also be 0, to leading order, or a two-circuit homoclinic orbit with initial conditions in I +2 (M) I 2 (M). Moreover, not only is p 0 = 0 to leading order, but p 0 0 exactly, since the reversibility symmetry (2.3) implies that the homoclinic orbits with initial conditions p 0 = 0inI +2 (M) I 2 (M)are also symmetric, just as we saw or the one-circuit orbits. That is, we have shown (2.26) I +2 (M) I 2 (M) {p=0}exactly. Finally, the same argument may be repeated inductively to show that (2.27) I +n (M) I n (M) {p=0} or all n =O(1)exactly.

18 460 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER It remains to determine, or which pairs m n and m + n even, whether or not (2.28) I +n (M) {p=0} and I m (M) {p=0}, so that we also have nonsymmetric multi-circuit homoclinic orbits to M. As shown above, only those orbits with initial conditions in I +1 (M) {p>0}can build I +2 (M). The main question then is: which o those orbits satisy p(ξ) = 0 at their second (or higher-order) intersection with {q = 0}? We begin by calculating the change in p during the hal-circuit rom {q = 0} back to itsel. Consider the orbit γ +1 (ξ) = (u +1 (ξ), p +1 (ξ), v +1 (ξ), q +1 (ξ)) with γ +1 (0) = (u +1 0,p+1 0,v+1 0,0) I+1 (M),wherenowp 0 +1 =ε p 0 >0with p 0 strictly O(1). Let Ξ (= O( log ε )) besuchthatγ +1 (Ξ) I +2 (M) {q=0}. We deine p by p +1 (Ξ) de = ε p 0 + p. Hence, by the second component o (2.2), Ξ (2.29) p +1 (u 0,ε p 0 )= ε [h 1 u α 1 v β 1 + ˆεu α 1 v β 1 H 1 (u, v; ε))] dξ 0 +O(ε 3 log ε ), and we see that p +1 (u 0,ε p 0 ) is inite, since β 1, β 2 > 0. Using the approximation ṽ h (ξ) deined by (2.14), we ind (2.30) p +1 (u 0,ε p 0 )= εh 1 u α (ṽ h(ξ)) β 1 dξ +O(ε 1+σ ), or some σ>0. Note that we can replace ṽ h (ξ) by v h (ξ) 0, the corresponding homoclinic solution to (2.7). The p-coordinate o γ +1 (Ξ) I +2 (M)is given to leading order by (2.31) [ ] ε p 0 h 1 u α 1 0 (v h (ξ)) β 1 dξ, with p 0 > 0. 0 Since u 0 > 0andv h (ξ) > 0, this expression can only change sign when (2.32) h 1 > 0, as assumed in (2.5). We see again thereore that, or h 1 > 0, all intersections I +n (M) I n (M)exist and satisy (2.27). Moreover, by ollowing the ast low or j hal circuits, we see that all I n+j (M) I n+j (M)exist, although these sets are not subsets o {p = 0},since p 0, by (2.30). Summarizing, (2.33) I +n (M) I m (M), or all n + m even, and m, n =O(1).

19 Large Stable Pulse Solutions in Reaction-diusion Equations 461 Remark 2.7. The condition β 1 > 0 in the hypotheses (2.5) o Theorem 2.1, which was imposed to help establish the existence o M, has been crucial here or a dierent reason. The quantity p +1 becomes unbounded as p 0 0 when β 1 < 0 (recall that the v-coordinate o γ +1 (Ξ) is at most O( ε)). In addition, the case β 1 = 0 is special, and we do not consider the details o this degenerate case here. Remark 2.8. We have also just seen the reason or imposing the requirement (2.32) in Theorem 2.1 (recall (2.5)). In the opposite case when h 1 < 0, the p- coordinate o γ +1 (Ξ) is strictly positive. Thus, I +2 (M) {p>εp +2 }or some 0 <p +2 =O(1). Analogously, it ollows that I 2 (M) {p< εp 2 } or some 0 <p 2 =O(1), and hence that I+2 (M) I 2 (M) =. Then, by repeating this argument, one readily sees also that (2.34) I +n (M) I n (M)=, or n>1, when h 1 < 0. In other words, i there exist homoclinic orbits to M when h 1 < 0 (with nonnegative u and v coordinates), then they can only be o the type that make at most one circuit through the ast ield Take o and touch down curves and the proo o Theorem 2.1. So ar, we have ocused on the dynamics in the ast ield o (2.2), and or every O(1) N>0, we have constructed a one-parameter amily o multi-circuit orbits homoclinic to M, under the conditions in the hypotheses (2.5) o Theorem 2.1. We now turn our attention or each N 1 to locating special pairs o curves on M that are essential or determining the slow segments o these same multi-circuit orbits. In particular, we will need the ideas developed in [11]. Let γ N (ξ) be an N-circuit orbit homoclinic to M, o the type whose existence has been shown in the previous subsection, with γ N (0) I N (M) I N (M) {p =0}. By geometrical singular perturbation theory (see [11] and [21]), there are two orbits γ M +N = γ+n M (ξ; (u+n 0,p 0 +N )) Mand γm N (ξ; (u N 0,p0 N )) M, respectively (where γ M ±N (0; (u±n 0,p 0 ±N )) = (u ±N 0,p 0 ±N ) M), such that γ N (ξ) γ M +N (ξ; (u+n 0,p 0 +N )) is exponentially small or ξ>0withξ O(1/ε) and γ N (ξ) γm N (ξ; (u N 0,p0 N )) is exponentially small or ξ<0with ξ O(1/ε). As a consequence, (2.35) d(γ N (ξ), M) =O(e k/ε ) or ξ O(1/ε) or larger, or some k>0. The orbits γ M ±N (ξ; (u±n 0,p 0 ±N )) determine the behavior o γ N (ξ) near M. Moreover,γ N (ξ) satisies the reversibility symmetry (2.3) by the choice o initial conditions, and thus (2.36) γm N (ξ; (u N 0,p N 0 )) = γ +N M ( ξ; (u+n 0, p +N 0 )).

20 462 ARJEN DOELMAN, ROBERT A. GARDNER & TASSO J. KAPER We now deine the curves T N d M( touch down ) and T N o M( take o ) as (2.37) T N d = γ N (0) T N o = γ N (0) {(u +N 0,p +N 0 )=γ N M {(u +N 0, p 0 +N )}, (0; (u+n 0,p +N 0 ))}, where the unions are over all γ N (0) I N (M) I N (M) {p=0} {q=0}. For each N = 1, 2,..., the take o set To N (respectively, the touch down set T N)is d the collection o base points o all o the Fenichel ibers in W U (M) (respectively, W S (M)) that have points in the transverse intersection o W U (M) and W S (M). Detailed asymptotic inormation about the locations o To N and T N d can be obtained explicitly by determining the relations between γ N (0) = (u 0, 0,v 0,0) and (u +N 0,p 0 +N,0,0). First, we observe that ṗ =O(ε 3 ) on M (2.2), thus, the p-coordinate o γ M +N remains constant to leading order during the ast excursions o γ N (ξ). Thereore, p 0 +N is completely determined (to leading order) by the accumulated change in p o γ N (ξ) during its time ξ>0inthe ast ield. These changes have already been calculated or N = 1 in (2.29) and (2.30). For N>1, the calculation is exactly the same, except or the act that γ N (ξ) now makes N hal circuits beore touching down on M: (2.38) p 0 +N = εh 1 Nu α 1 0 (v h(ξ)) β 1 dξ +O(ε 1+σ ), 0 or some σ>0. From the irst component o (2.2) and the act that p =O(ε), we also conclude that u +N 0 = u 0 to leading order. Thus, we ind (2.39) { } T N : d p de = p N d (u) = εh 1Nu α 1 (v h(ξ)) β 1 dξ +O(ε 1+σ ), 0 T N o : {p = p N d (u)} (σ >0), where v h (ξ) = v h (ξ; u), the homoclinic solution o (2.7). To more ully determine the behavior o p N d as a unction o u, weintroduce w h = w h (ξ; β 2 ) 0, which is the (positive) homoclinic solution o a rescaled version o (2.7): (2.40) ẅ = w w β 2. Without loss o generality, we take the solutions to be parameterized such that w h (ξ) is symmetric with respect to ξ = 0. Thus, (2.41) v h (ξ; u) = v h (ξ; u, h 2,α 2,β 2 )=(h 2 u α 2 ) 1/(1 β 2) w h (ξ; β 2 ),

21 which yields Large Stable Pulse Solutions in Reaction-diusion Equations 463 (2.42) (v h (ξ)) β 1 dξ = 2 (v h (ξ)) β 1 dξ 0 = h β 1/(β 2 1) 2 u α 2β 1 /(β 2 1) W(β 1,β 2 ), where (2.43) W(β 1,β 2 )= (w h (ξ; β 2 )) β 1 dξ. We can now rewrite (2.39) to leading order as (2.44) T N d,o : {p =±pn d (u)}, with p N d (u) = N 2 εh 1h β 1/(β 2 1) 2 u 1+D/(β 2 1) W(β 1,β 2 ), with D>0 (2.10). Note that the higher order corrections, which are not needed here, can be obtained by a straightorward asymptotic approximation scheme, see [9]. In Figure 2.1 on the ollowing page, we have plotted the curves T N d,o or a ew values o N superimposed onto the linear low on M given by (2.12). Since D>0 and β 2 > 1, the curve To N given by (2.44) to leading order is tangent to the u-axis or each N; and, thus, by (2.13), To N lu exists or all N. In act, the unique intersection is given to leading order by (2.45) u h,n = 2hβ 1/(β 2 1) 2 µ h 1 NW(β 1,β 2 ) (β 2 1)/D. Then, the reversibility symmetry (2.3) implies that T N d ls also, with precisely the same value o u given by (2.45). Thereore, or each N>0, there is a unique homoclinic orbit to the saddle point S = (0, 0, 0, 0) in (2.2) that (i) lows outward rom S staying exponentially close to l U (and hence also to M) untiltheu coordinate reaches a neighborhood o u h,n, (ii) makes N ull circuits through the ast ield near the homoclinic orbit (v h (ξ), q h (ξ)) o (2.7) while u is constant (= u h,n ), and (iii) returns to an exponentially small neighborhood o M, lowing in toward S along l S. We have shown in Sections 2.2 (N = 1) and 2.3 (N >1) that this homoclinic orbit lies in the transverse intersection o the maniolds W U (M) and W S (M). Thereore, we can conclude that there exists an orbit γ N h (ξ) = (un h (ξ), pn h (ξ), v N h (ξ), qn h (ξ)) W U (M) W S (M) that is asymptotically close to an orbit on l U Mor ξ 1/ε and asymptotically close to an orbit on l S Mor