Numerical Computation of Solitary Waves on Innite. Cylinders

Transcription

1 Numerical Computation of Solitary Waves on Innite Cylinders Gabriel J. Lord CISE National Physical Laboratory Queens Road Teddington TW11 LW, U.K. Daniela Peterhof Weierstra-Institut fur Angewandte Analysis und Stochastik Mohrenstrae Berlin, Germany Bjorn Sandstede Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH , USA Arnd Scheel Institut fur Mathematik I Freie Universitat Berlin Arnimallee Berlin, Germany Abstract The numerical computation of solitary waves to semilinear elliptic equations in innite cylinders is investigated. Rather than solving on the innite cylinder, the equation is approximated by a boundary-value problem on a nite cylinder. Convergence and stability results for this algorithm are given. In addition, it is shown that Galerkin approximations can be used to calculate solitary waves for the elliptic problem on the nite cylinder. The theoretical predictions are compared with numerical computations. In particular, post buckling of an innitely long cylindrical shell under axial compression is considered; it is shown numerically that, for a xed spatial truncation, the error in the truncation on the length of the cylinder scales in accordance with the theoretical predictions. Keywords: solitary wave, boundary-value problem, elliptic equation AMS subject classication: 65N12, 35B3, 35J55, 65N3

2 1 Introduction The numerical computation of solitary-wave solutions to elliptic systems u xx + y u + g(y; u; u x ; r y u) = ; (x; y) 2 R ; (1.1) in innite cylinders R is investigated. Here, u 2 R m, and is an open and bounded subset of R n with Lipschitz boundary. Appropriate boundary conditions R((u; u x ; r y u)j R@ ) = (1.2) on should be added. Solitary waves are solutions h(x; y) satisfying lim h(x; y) = p (y) x!1 uniformly for y 2. In applications, they frequently arise as travelling waves h(x ct; y) for parabolic equations u t = u xx + y u + g(y; u; u x ; r y u); (x; y) 2 R : (1.3) These applications include problems in structural mechanics such as shells and struts, chemical kinetics, combustion, and nerve impulses; see, for instance, [32] and the comprehensive bibliography there. Analytically, the existence of solitary-wave solutions exhibiting non-trivial structure in the direction of the cross-section is still a largely open problem. Existence has been proved in many cases for small solutions using center-manifold theory [19, 25]. In special cases, it may be possible to exploit maximum principles [1, 2, 17] and variational structure [27, 28]. Another approach establishing existence of front solutions uses topological methods [1, 12]. Suppose that h(x; y) is a solitary wave which satises (1.1{1.2). In order to calculate h numerically, the problem on the innite cylinder R has to be approximated by a suitable system u xx + y u + g(y; u; u x ; r y u) = ; (x; y) 2 (T ; T + ) (1.4) R((u; u x ; r y u)j [T ;T+]@) = ; posed on a nite cylinder. Here, we have to specify appropriate conditions R ((u; u x ; r y u)j ft g ) = ; (1.5) R + ((u; u x ; r y u)j ft+g) = ; at the boundaries induced by the truncation of the cylinder axis. The issue is then to determine whether equation (1.4{1.5) has a unique solution close to the solitary wave h, and if it does, to derive estimates for the error caused by the truncation. 1

3 In this article, we give sucient conditions on the equation and the boundary conditions (1.5) such that the aforementioned algorithm works. Boundary conditions which satisfy these conditions are called admissible. One implication of our assumptions is that the solitary wave h(x; y) converges exponentially towards p (y) as jxj! 1 uniformly in y 2. The dierence of the solution h T of (1.4{1.5) and the solitary wave h can then be estimated by jh h T j C(jR (hj ft g )j + jr + (hj ft+g)j); in appropriate norms, where the positive constant C does not depend on T and T +. Here, the right-hand side converges to zero exponentially as jt j; T +! 1. In order to prove this result, we interpret the variable x as time, and write (1.1) as a rst-order system! ux = v x! v : (1.6) y u g(y; u; v; r y u) Here, for each xed x 2 R, (u; v)(x) is a function of y 2 contained in some function space depending on the boundary conditions A solitary wave of (1.1) corresponds to a homoclinic or heteroclinic solution of (1.6) which connects the equilibria p (y) and p + (y), that is, we have lim (h(x); h x(x))! (p ; ) (1.7) x!1 in the underlying function space. We then investigate the truncated boundary-value problem by replacing (1.7) by a condition of the form R ((u; v)(t )) = ; R + ((u; v)(t + )) = : (1.8) The key for solving the resulting boundary-value problem are exponential dichotomies for the linearization! u x = v id y D u g D ryug r y D ux g of (1.6) about the solitary wave (h(x); h x (x)). 1! A u ; (1.9) v Here, derivatives of g are evaluated at (y; h; h x ; r y h). Exponential dichotomies are projections onto x-dependent stable and unstable subspaces, say E s (x) and E u (x), such that solutions (u; v)(x) of (1.9) associated with initial values (u; v)(x ) in the stable space E s (x ) exist for x > x and decay exponentially for x! 1. In contrast, solutions (u; v)(x) associated with initial values (u; v)(x ) in the unstable space E u (x ) satisfy (1.9) in backward x-direction x < x and decay exponentially for decreasing x. In the context of elliptic equations, the stable and unstable spaces are both innite-dimensional. Existence of exponential dichotomies for ordinary, parabolic 2

4 or functional dierential equations is well known. For elliptic equations, existence has recently been proved in [26] using a novel functional-analytic approach. The results in this latter article then allow us to solve the truncated boundary-value problem and to derive the aforementioned error estimate. It remains to actually solve the truncated boundary-value problem. There are two dierent ways of accomplishing this task. Firstly, we concentrate on the elliptic formulation. Consider, for instance, equation (1.1) with Dirichlet boundary conditions, that is, u xx + y u + g(y; u; u x ; r y u) = ; (x; y) 2 R uj R@ = ; and assume that the solitary wave converges to zero as jxj tends to innity. We may then want to take Dirichlet boundary conditions for the articial conditions (1.5) which results in the truncated problem u xx + y u + g(y; u; u x ; r y u) = ; (x; y) 2 (T ; T + ) ;T+)) = : This system can now be discretized using nite dierences or nite elements. Of course, the same procedure works for Neumann or periodic boundary conditions provided they are admissible. Secondly, we could take a dynamical-systems point of view and consider the rst-order system ( ). We now discretize only in the cross-section and obtain a large system of ODEs! u x v x! v = Q n ; y u g(y; u; v; r y u) = R ((u; v)(t )); = R + ((u; v)(t + )); dened on R(Q n ) where the Galerkin projection Q n projects the function space in onto a nite-dimensional subspace. ODE codes such as Homcont, see [7, 9], can then be used to solve the resulting boundary-value problem. Often, elliptic equations have an additional reection symmetry. For instance, consider the fourth-order equation u xxxx + 2 yu + g(y; u; u xx ; y u) = ; (x; y) 2 R ; (1.1) which is included in our general set-up. The Z 2 -symmetry u(x; y) 7! u( x; y) leaves (1.1) invariant. This symmetry manifests itself as a time-reversibility S : (u; v 1 ; v 2 ; v 3 ) 7! (u; v 1 ; v 2 ; v 3 ) for the associated dynamical system (u; v 1 ; v 2 ; v 3 ) x = (v 1 ; v 2 ; v 3 ; 2 y u g(y; u; v 2; y u)): 3

5 It is shown that the algorithm given above can be adapted to this situation. We also remark that our results apply to parabolic equations u t = u + g(u; ru); x 2 ; with R n bounded and open. Here, we are interested in the computation of homoclinic or heteroclinic solutions h(t; x) which satisfy lim t!1 h(t; x) = p (x). For ordinary dierential equations, well-posedness of the truncated problem has been investigated by Beyn [4], and Doedel and Friedman [11] for very general boundary conditions. In addition, error estimates have been derived in these articles. Hagstrom and Keller [15, 16] considered elliptic problems of the form (1.1) assuming that (1.9) has an exponential dichotomy. They investigated the so-called asymptotic boundary conditions. These conditions select precisely those solutions converging to p (y) as x! 1. In particular, the solution of the truncated problem coincides with the true wave h on the innite cylinder. The actual calculation of the asymptotic boundary conditions, however, involves again certain approximations which were not investigated in [15, 16]. As a concrete application, we consider the post buckling of an innitely long cylindrical shell under axial compression as modeled by the von Karman{Donnell equations. In [22, 23] and [24] solitary-waves were computed as solutions representing localized buckling patterns and it was shown that these solutions provide a good approximation to the localized buckling pattern observed in experiments on long shells. The numerical procedure involved the reduction to a truncated boundary-value problem and its discretization using Galerkin approximation as discussed above. Here, we show numerically that, for a xed spatial truncation, the error in the truncation on the length of the cylinder scales in accordance with our theoretical predictions. This paper is organized as follows. Section 2 contains the general set-up and the main results. We summarize the results about exponential dichotomies from [26] in Section 3. The theorems on Galerkin approximations and the truncated boundary-value problem are proved in Sections 4{6. We show in Section 7 that our results apply to semilinear elliptic equations. Numerical simulations and a comparison of the numerical and theoretical error are presented in Section 8. Finally, the application to the von Karman{Donnell equation is given in Section 9. Acknowledgements BS is grateful to Andrew Poje for helpful discussions on discretization issues and for allowing us to use his conjugated-gradient solver. DP was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant Schn426/5-1. GJL was supported by the EPSRC, UK. 4

6 2 Main Results 2.1 The Setting Assume that A is a closed operator dened on a reexive Banach space X with dense domain D(A). Let B 2 L(X) be any bounded operator. We say that A and B commute if Bu 2 D(A) for any u 2 D(A) and AB = BA on D(A). Hypothesis (A1) Suppose that there is a constant C such that k(a i) 1 k L(X) C 1 + jj for all 2 R. Assume that there is a projection ^P 2 L(X) with the following properties: A 1 and ^P commute, and there exists a > such that Re < for any 2 spec(a ^P ) and Re > for any 2 spec(a(id ^P )). Throughout, C denotes various dierent constants all independent of T and T +. Sucient conditions for the existence of the projection ^P, which is sometimes referred to as the Calderon projector [6], have been given in [5, 13]. We also refer to the explicit construction of the projections for semilinear elliptic equations in [26, Section 6]. Dene ^P+ = id ^P and A = ^P A, A + = ^P+ A, and let X = R( ^P ) and X + = R( ^P+ ). Here, range and kernel of an operator L are denoted by R(L) and N(L), respectively. By Hypothesis (A1), the operators A and A + are sectorial with their spectrum contained in the right half-plane. Therefore, for, we can dene the interpolation spaces X = + D(A +) and X = D(A ), see [18]. Finally, we set X = X + X. We denote the norm in X by j j and the operator norm in L(X ) by k k. The projection ^P is then in L(X ) for any < 1. In addition, we assume that A has compact resolvent. Hypothesis (A2) The operator A 1 2 L(X) is compact. In the following, we consider the abstract evolution equation _u = Au + f(u; ); (u; ) 2 X R (2.1) for some xed 2 [; 1) and for f 2 C 2 (X R; X). We say that u(t) is a solution of (2.1) on the interval [; T ) if u 2 C 1 ((; T ); X) \ C ((; T ); D(A)) \ C ([; T ); X ) 5

7 and u(t) satises (2.1) in X for t 2 (; T ). equilibrium and of a homoclinic orbit for =. We assume the existence of a hyperbolic Hypothesis (H1) Equation (2.1) has a hyperbolic equilibrium p 2 D(A) for =. In particular, A + D u f(p ; ) meets Hypothesis (A1). Hypothesis (H2) Let h(t) 2 C 1 (R; X ) \ C (R; X 1 ) be a homoclinic solution of (2.1) for = with h(t)! p as jtj! 1. We assume that h(t) is non-degenerate, that is, h(t) _ is the only bounded solution, up to constant multiples, of the variational equation _v = Av + D u f(h(t); )v (2.2) about h(t). Next, we introduce the adjoint variational equation _v = (A + D u f(h(t); ) )v; (2.3) about the homoclinic solution h(t). To describe the asymptotic behavior of solutions of (2.2) and (2.3), we have to assume forward and backward uniqueness. Hypothesis (A3) The only bounded solution of (2.2) and (2.3) on R + or R with v() = is the trivial solution v(t) =. Hypotheses (H2) and (A3) imply that the adjoint equation (2.3) has a unique, up to scalar multiples, bounded solution (t) on R. Finally, we assume that the Melnikov integral associated with h(t) does not vanish. Hypothesis (H3) M := R 1 1 h (t); D f(h(t); )i dt 6= : 2.2 The Galerkin Approximation First, we show persistence of the homoclinic orbit h(t) under nite-dimensional Galerkin approximations of equation (2.1). We may think of a Galerkin approximation as a family of projections denoted by Q 2 L(X) for >. Here, Q = id, while Q will have nite-dimensional range for > and approximates the identity in a weak sense. Hypothesis (Q) (i) A commutes with Q. 6

8 (ii) The norms kq k L(X) C are bounded uniformly in. (iii) For any u 2 X, we have jq u uj! as!. It is a consequence of Hypothesis (Q)(i) that A Q = Q A. Therefore, we have Q 2 L(X ) and kq k L(X ) C independently of >. Furthermore, jq u uj! as! for any u 2 X. In order to obtain uniform convergence of the Galerkin approximation, we assume compactness of the nonlinearity f. Hypothesis (C) compact map. If Q 6= id for some >, we assume that f : X R! X is a The next theorem shows the persistence of the equilibrium and the homoclinic orbit under the Galerkin approximation _u = Au + Q f(u; ); (u; ) 2 X R; (2.4) of equation (2.1). We emphasize that the subspaces R(Q ) and N(Q ) are both invariant under equation (2.4). For initial data u 2 (id Q )X, equation (2.4) reduces to the linear equation _u = Au, which has no bounded solution on R except u =. We also remark that the norms on Q X and Q X are equivalent, but the equivalence constants tend to innity as!. Therefore, estimates which are uniform with respect to can only be expected in the X -norm. Theorem 1 Assume that Hypotheses (A1){(A3), (H1){(H3), (C), and (Q) are satised. There are then positive numbers,, and C such that the following is true for any < and jj <. (i) Equation (2.4) has a hyperbolic equilibrium p () 2 R(Q ) with p () = p and jp () p j C(j(id Q )p j + jj): (ii) For every, there exists a such that equation (2.4) has a non-degenerate homoclinic orbit h (t) 2 Q X with h (t)! p ( ) as jtj! 1, and j j + sup t2r jh (t) h(t)j C sup j(id Q )h(t)j : t2r (iii) Besides p () and h (t), there are no other equilibria or homoclinic solutions of (2.4) in the open set f(u; ) 2 X R; jj + inf t2r ju h(t)j < g. 7

9 We denote the spectral projections associated with A+D u f(p ; ) and A+Q D u f(p (); ) in X by P and P ; (), respectively. In particular, P and P ; () project onto the stable eigenspaces corresponding to eigenvalues with negative real part of A + D u f(p ; ) and A + Q D u f(p (); ), respectively. 2.3 The Truncated Boundary-Value Problem Here, we investigate the numerical computation of the homoclinic orbits h of the Galerkin approximation. The approach most commonly used consists of truncating the innite interval R to a nite interval [T ; T + ] for some T < < T + and imposing boundary conditions at the end points t = T and t = T +. The truncated boundary-value problem is given by B _u Au Q f(u; ) R (u(t + ); u(t ); ) J T; (u; ) 1 C C A = (2.5) for t 2 T = (T ; T + ). Here, J T; denotes a phase condition and R encodes the boundary conditions. They have to satisfy the following conditions. Hypothesis (T1) (i) J T; : C (T; X ) R! R is of class C 2, and J T; (h ; )! as jt j, T +! 1. Furthermore, there is a constant d > independent of T, T + and such that D u J T; (h ; ) h _ d > for all jt j, T + suciently large. Finally, D u J T; (u; ) and DuJ 2 T; (u; ) are bounded in a ball in C (T; X ) R of xed radius centered at (h ; ) uniformly in T, T +, and. (ii) We have R 2 C 2 (X X R; X ) such that DR and D 2 R are bounded in a small ball centered at (p ( ); p ( ); ) in X X R uniformly in. Furthermore, D u+;u R (p ( ); p ( ); )j R(P+;( ))R(P ;( )) is invertible, and the inverse is bounded uniformly in. Note that _ h() and _ h () are contained in C (T; X ). Therefore, the condition on J in (T1)(i) makes sense. Remark 2.1 The boundary conditions are often separated, that is, given by R (u + ; u ; ) = (R +; (u + ; ); R ; (u ; )) 2 R(P +; ( )) R(P ; ( )) = X : 8

10 If the operators D u R ; (p ( ); )j R(P;( )) are invertible, and the inverses are bounded uniformly in, then the invertibility condition in Hypothesis (T1)(ii) is also satised. We have the following theorem. Theorem 2 Assume that (A1){(A3), (H1){(H3), (C), (Q) and (T1) are met. There exist positive numbers, and C such that for all suciently large intervals T the following is true. For any 2 [; ), the boundary-value problem (2.5) has a unique solution ( h (t); ) in the tube and f(u; ) 2 C ([T ; T + ]; X ) R; jj + sup ju(t) h(t)j g; t2[t ;T +] j j + sup j h (t) h (t + T; )j CjR (h (T + ); h (T ); )j t2[t ;T +] for an appropriate small number T;. Combining Theorems 1 and 2, and exploiting Hypothesis (T1), we obtain the following corollary. Corollary 1 Under the assumptions of Theorem 2, we have the estimate j j + sup t2[t ;T +] j h (t) h(t)j j C jr (h(t + ); h(t ); )j + sup t2r j(id Q )h(t)j for the dierence of the true homoclinic orbit h and the numerical approximation h obtained by solving the boundary-value problem on a nite interval for the Galerkin approximation of (2.1). The error estimate can be made more explicit. Corollary 2 Under the assumptions of Theorem 2, we have the estimate j j + sup t2[t ;T +] j h (t) h(t)j C(e s T + + e ut + sup j(id Q )h(t)j ); t2r where the constants s and u are chosen such that =2 spec(a+d u f(p ; )) for any 2 C with Re 2 [ s ; u ]. We point out that the case Q = id for all is included in the analysis. It corresponds to truncating equation (2.1) directly without going to a nite-dimensional approximation. A more analytical consequence of Theorem 2 is the existence of periodic solutions with large period near the homoclinic orbit h. 9

11 Corollary 3 Assume that (A1){(A3) and (H1){(H3) are met. There is then a constant > such that (2.1) has a periodic orbit (u ; ) with minimal period for any >. Furthermore, j j + sup t2[ 1 2 ; 1 2 ] ju (t) h(t)j C(e s + e u ); where the constants s and u are as in Corollary 2. Proof. Consider the phase condition J T (u) := h'; u()i where ' 2 (X ) is chosen such that h'; h()i _ = 1. The boundary condition is R(u + ; u ; ) = u + u. Since R(P + ) R(P ) = X, Hypothesis (T1) is satised, and we can apply Theorem 2 with Q = id for all. This corollary has been proved for ordinary dierential equations in [3] and [21]. The proof given in [21] also covers parabolic and functional-dierential equations. Our contribution is the extension to elliptic equations. 2.4 The Algorithm in Practice In practice, the Galerkin approximation is considered on the nite-dimensional space R(Q ), that is, _q = Aq + Q f(q; ); (q; ) 2 R(Q ) R: (2.6) If X is a Hilbert space, the phase condition may be chosen according to J T; (q; ) = Z T+ T h _ h (t); q(t) h (t)i X dt: (2.7) For the boundary conditions, we may take, for instance, the projection boundary conditions which are dened by R +; (q(t + ); ) = Q +; ()(q(t + ) p ()); (2.8) R ; (q(t ); ) = Q ; ()(q(t ) p ()); where Q +; () and Q +; () are the unstable and stable spectral projections in R(Q ) of the operator (A + Q D u f(p (); ))j R(Q). We have the following result for the nitedimensional boundary-value problem on R(Q ) described above. Theorem 3 Assume that (A1){(A3), (H1){(H3) and (C) are met where X is a Hilbert space. There exist positive numbers,, and C such that for all suciently large intervals 1

12 T the following is true. For any 2 [; ), the boundary-value problem (2.6{2.8) on R(Q ) has a unique solution ( h (t); ) in the tube and j j + f(q; ) 2 C ([T ; T + ]; R(Q )) R; jj + sup jq(t) h(t)j g; t2[t ;T +] sup j h (t) h(t)j C(e 2s T + + e 2uT + sup j(id Q )h(t)j ); t2[t ;T +] t2r for numbers s and u as in Corollary 2. We point out that super-convergence in the parameter occurs. Indeed, following the proof given in [29], we have j j C(e (2s + u )T + + e (2u + s )T + sup j(id Q )h(t)j ): t2r 2.5 Reversible Systems In applications, elliptic equations are often time-reversible. Here, we account for this property, and adapt the algorithms described above to reversible systems. Consider equation (2.1) _u = Au + f(u); u 2 X : (2.9) Time-reversibility is encoded in the following hypothesis. Hypothesis (R) Suppose that S 2 L(X) is a bounded operator such that (i) S anti-commutes with A and f, that is, SA = AS and f(su) = Sf(u) on D(A). (ii) S 2 = id. (iii) S commutes with Q for all. We remark that S 2 L(X ) on account of (R)(i). Finally, we assume that the homoclinic solution h(t) is symmetric and a certain transversality condition is satised. Hypothesis (H4) (i) Sh() = h(), that is, h() 2 Fix(S). (ii) Fix(S) R( s +(; )) = X. 11

13 Here, s +(t; ) denotes the stable evolution of the variational equation (2.2) about h(t); see Theorem 5 below. We then solve the boundary-value problem _u = Au + Q f(u); (id S)u() = ; R +; (u(t + )) = ; (2.1) on the interval [; T + ]. Hypothesis (T2) Suppose that ^X are closed subspaces of X. Assume that R +; 2 C 2 (X R; ^X ) such that DR and D 2 R are bounded in a small ball centered at (p ( ); ) in X R uniformly in. Furthermore, D u R +; (p ( ); )j R(P+;( )) is invertible, and the inverse is bounded uniformly in. Theorem 4 Assume that (A1){(A3), (H1){(H2), (H4), (C), (Q) and (R) are met. Suppose that R +; satises (T2). The boundary-value problem (2.1) has then a unique solution h for all T + suciently large and small enough. Furthermore, h () 2 Fix(S) is symmetric, and sup j h (t) h(t)j C(jR +; (h(t + ))j + sup j(id Q )h(t)j ): t2[;t +] t2r The statements of Theorem 1 and 3 are also true for (2.1) if adapted appropriately. Corollary 4 Assume that (A1){(A3), (H1){(H2), and (R)(i) and (ii) are met. There is then a constant > such that (2.9) has a periodic orbit u any >. Furthermore, where the constant s is as in Corollary 2. sup t2[ 1 2 ; 1 2 ] ju (t) h(t)j Ce s ; with minimal period for Proof. We apply Theorem 4 with R + (u) = 1(id S)u and ^X 2 = R(id S). By Dunford-Taylor calculus and (R)(i), we have SP + = P S. Moreover, due to (R)(ii), v 2 R(id S) implies Sv = v. Using these facts, it is then straightforward to show that P + (id S)j R(P+) = id R(P+) and (id S)P + j R(id S) = id R(id S). Hence, Hypothesis (T2) is satised. It follows as in [31] that the solution of this boundary-value problem is periodic. Note that s = u due to reversibility. 2.6 Computation of Heteroclinic Orbits We emphasize that the results presented thus far also apply to heteroclinic orbits, that is, solutions connecting two dierent equilibria p as t! 1. Here, we briey outline 12

14 the necessary changes. Suppose that p are hyperbolic equilibria of (2.1) which satisfy Hypothesis (H1). Furthermore, assume that h(t) satises (H2) but with lim t!1 h(t) = p. In particular, (H2) implies that the heteroclinic orbit h(t) is isolated. Next, we assume that (A1){(A3) are met. As a consequence of (H2), (A3) and [26, Corollary 1], the adjoint variational equation (2.3) _v = (A + D u f(h(t); ) )v about the heteroclinic orbit h(t) has only nitely many, linearly independent bounded solutions j (t) for j = 1; :::; m on R. Hypothesis (H3) is then replaced by the following assumption. Hypothesis (H5) Let 2 R m and assume that the m m matrix M with entries M ij := R 1 1 h i (t); D j f(h(t); )i dt is invertible. Note that Hypothesis (H5) is automatically met if m =, that is, if the heteroclinic orbit is transversely constructed. With Hypothesis (H3) replaced by (H5), Theorem 1 remains true. Let p ; () denote the perturbed equilibria for (2.4). We denote the spectral projections of A + D u f(p ; ) and A + Q D u f(p ; (); ) onto the stable eigenspaces corresponding to eigenvalues with negative real part by P and P ; (), respectively. Similarly, P + and P +; () are the spectral projections of A + D u f(p + ; ) and A + Q D u f(p +; (); ), respectively, onto the eigenspaces corresponding to eigenvalues with positive real part. Suppose that the boundary conditions are given by R (u + ; u ; ) = (R +; (u + ; ); R ; (u ; )) 2 R(P +; ( )) R(P ; ( )): We assume that D u R ; (p ; ( ); )j R(P;( )) is invertible, and the inverse is bounded uniformly in. If Hypothesis (T1)(ii) is replaced by this assumption, the results in the previous sections remain true. 3 Exponential Dichotomies { an Excursion Here, we summarize the results in [26] which are the key to the proofs of the theorems presented in the last section. Assume that the operator A is as in Section 2, that is, A : D(A) X! X is a closed operator such that its domain D(A) is dense in X. Furthermore, A satises Hypotheses (A1) and (A2). Moreover, let B 2 C (R; L(X ; X)) be a continuous family of operators. 13

15 Consider the dierential equation _v = (A + B(t)) v: (3.1) We are particularly interested in solutions v(t) with some prescribed exponential behavior for t 2 R + and t 2 R. Denition (Exponential Dichotomy) Equation (3.1) has an exponential dichotomy in X on the interval J R if there exist positive constants C and, and operators s (t; ) and u (; t) in L(X ) dened for t with t; 2 J such that the following is true. (i) For any v 2 X, s (t; )v is a solution of (3.1) for t in J. Similarly, u (t; )v is a solution of (3.1) for t in J. (ii) For any v 2 X, s (t; )v and u (; t)v are continuous in t in J. (iii) k s (t; )k + k u (; t)k Ce (t ) for all t in J. (iv) s (t; ) s (; s) = s (t; s) for all t s in J, and the analogous property for u (; t). Note that the operators P (t) = s (t; t) are projections. We assume that B(t) is small for large jtj. The constant is specied in Theorem 5 below. Hypothesis (D1) There are numbers # > and t > such that B 2 C ;# (R; L(X ; X)) and kb(t)k L(X ;X) for all jtj t. Finally, we assume forward and backward uniqueness of solutions of equation (3.1) on the interval R. Hypothesis (D2) The only bounded solution v(t) of (3.1) on the intervals R + or R with v() = is the trivial solution v(t) =. Similarly, the only bounded solution w(t) of the adjoint equation _w = (A + B(t)) w on R + or R with w() = is w(t) =. We then have the following existence result for exponential dichotomies of (3.1). Theorem 5 ([26]) Suppose that Hypotheses (A1){(A2) and (D2) are satised. There is then a constant > such that (3.1) has an exponential dichotomy on R + provided Hypothesis (D1) is met with =. 14

16 (i) The projections P (t) = s (t; t) are Holder continuous in t 2 R + with values in L(X ). (ii) The operator s (t; ) has a bounded extension to X satisfying s (t; ) s (; s) = s (t; s) for all t s. (iii) s (t; ) 2 L(X; X ) for t > and k s (t; )k L(X;X ) C(t ) e (t ). Analogous properties hold for u (; t) with t. The same results is true with R + replaced by R. We denote the evolution operators by s +(t; ) and u +(; t) for t, and by s (; t) and u (t; ) for t. Finally, we compare the evolution operators for two dierent equations. Lemma 3.1 ([26]) Suppose B 1 (t) and B 2 (t) satisfy the assumptions of Theorem 5 on J = R +. There exist positive numbers C and such that the following is true. If sup kb 1 (t) B 2 (t)k L(X ;X) < ; t the projections P j (t), j 2 f1; 2g, which correspond to the equations _v = (A + B j (t))v satisfy the estimate sup kp 1 (t) P 2 (t)k L(X ) C: t 4 The Galerkin Approximation In this section, Theorem 1 is proved. Throughout, C denotes various dierent constants all independent of T and T +. We will use the following version of Banach's xed point theorem. Lemma 4.1 Suppose that Y and ^Y are Banach spaces and G : Y! ^Y is a C 1 -function. Assume that there exists a linear, bounded and invertible operator L : Y! ^Y, an element y 2 Y, and numbers > and < < 1 such that (i) k id L 1 DG(y))k for all y 2 B (y ), (ii) jl 1 G(y )j (1 ). There exists then a unique point y 2 B (y ) with G(y ) = and jy y j (1 ) 1 jl 1 G(y )j; kdg(y) 1 k (1 + ) kl 1 k; uniformly in y 2 B (y ). 15

17 Here, B (y) is the ball with center y and radius in Y. We start with a useful consequence of Hypothesis (Q). Lemma 4.2 Assume that (Q) is satised and let K 2 L(X ; X) be a compact operator. Then k(id Q )Kk L(X ;X)! as!. Proof. We argue by contradiction. Suppose that there are elements v n 2 X and n > with jv n j = 1 and n! as n! 1 such that j(id Q n )Kv n j >. After choosing a subsequence, we have Kv n! w in X since K is compact. Hence, j(id Q n )Kv n j j(id Q n )wj + j(id Q n )(Kv n w)j j(id Q n )wj + CjKv n wj! ; as n! 1, due to (Q)(ii). This is a contradiction. Note that D u f(u; ) is compact for any (u; ) provided Hypothesis (C) is satised. 4.1 Persistence of the Equilibrium In order to show that the equilibrium persists, consider (A + D u f(p ; )) 1 (A(p + u) + Q f(p + u; )) = u + (A + D u f(p ; )) 1 Q (f(p + u; ) f(p ; ) D u f(p ; )u) +(id Q )(f(p ; ) + D u f(p ; )u) =: G (u; ): It suces to seek zeroes of G (u; ) near (p ; ). The map G is smooth in (u; ) as a map from X R to X and satises G (; ) = as well as D u G (; ) = id. Furthermore, using Ap + f(p ; ) =, jg (; )j Cj(A + D u f(p ; )) 1 (Q (f(p ; ) f(p ; )) + (id Q )f(p ; ))j C(jj + j(a + D u f(p ; )) 1 A(id Q )p j ) C(jj + j(id Q )p j ); and, due to (C) and Lemma 4.2, kd u G(u; ; ) id k CkD u f(p + u; ) D u f(p ; ) + (id Q )D u f(p ; )k < 1 2 for all (u; ; ) in a ball in X R 2 centered at the origin with suciently small radius. We apply Lemma 4.1 for any (; ) in B () R 2 with L = id. Hence, there exists a unique zero p () 2 B (p ) X of G (; ), with p () = p and jp () p j C(j(id Q )p j + jj): 16

18 Furthermore, p () is smooth in. By construction, p () are equilibria of (2.4). By Hypothesis (C), Theorem 5, and Lemma 4.2, the equation _v = (A + Q D u f(p (); ))v has an exponential dichotomy on R with projections P +; () and P ; (). This proves the rst, and part of the third claim in Theorem Persistence of the Homoclinic Orbit Next, we introduce a new variable v by u(t) = h(t) + v(t); (4.1) and write equation (2.4), that is _u = Au + Q f(u; ), in the form _v = (A + D u f(h(t); ))v + F (t; v; ) (4.2) = (A + D u f(h(t); ))v + D f(h(t); ) + ^F (t; v; ); with ^F (t; v; ) := (id Q ) D u f(h(t); )v + D f(h(t); ) + f(h(t); ) +Q f(h(t) + v; ) f(h(t); ) D u f(h(t); )v D f(h(t); ) Due to Hypothesis (C) and Lemma 4.2, we have the estimate kd (u;) ^F (t; v; )k L(X ;X) C(jvj + jj) + g(); (4.3) for some function g() with g()! as!. On account of Theorem 5 and Hypotheses (A1){(A3), we know that equation (2.2) _v = (A + D u f(h(t); ))v has exponential dichotomies on R + and R. As in Theorem 5, we denote the solution operators of this equation by s +(t; ) and u +(; t) for t, and by s (; t) and u (t; ) for t. Solutions of the nonlinear equation (4.2) are bounded on R if, and only if, there exist (b + ; b ) 2 R( s +(; )) R( u (; )) such that v + (t) = s (t; )b v (t) = u (t; )b + v + () = v () = h'; v + ()i: Z t Z t s + (t; )F (; v + (); ) d + u (t; )F (; v (); ) d + 17 Z t 1 Z t 1 u + (t; )F (; v + (); ) d; s (t; )F (; v (); ) d; :

19 Here, ' 2 (X ) is chosen such that h'; _ h()i = 1. The last equation takes care of the translational invariance of (2.4). In the rst and second equation, we have t 2 R + t 2 R, respectively. We remark that it suces to seek weak solutions of (4.2) since any weak solution is actually a strong solution, see [26, Lemma 3.1]. Let G (b + ; b ; v + ; v ; ) := B R R v + (t) s t +(t; )b + s t R +(t; )F (; v + (); ) d R 1 u +(t; )F (; v + (); ) d v (t) u (t; )b R t u t (t; )F (; v (); ) d R 1 s (t; )F (; v (); ) d 1 b + b u (; )F + (; v + R (); ) d 1 s (; )F (; v (); ) d h'; s 1 +(; )b + u +(; )F (; v + (); ) di and consider G : Y! ^Y for xed as a map dened on the spaces Y := R( s +(; )) R( u (; )) C (R + ; X ) C (R ; X ) R; ^Y := C (R + ; X ) C (R ; X ) X R: Note that G is well dened and smooth in (b + ; b ; v + ; v ; ). We exploit the splitting (4.2) of F into the linear term D f(h(t); ) and the quadratic term ^F, see (4.3). Therefore, consider G (b + ; b ; v + ; v ; ) = L(b + ; b ; v + ; v ; ) R t s + (t; ) ^F (; v + (); ) d + R t 1 u + (t; ) ^F (; v + (); ) d R t u (t; ) ^F (; v (); ) d + R t 1 s (t; ) ^F (; v (); ) d R 1 u +(; ) ^F (; v + (); ) d + R 1 s (; ) ^F (; v (); ) d h'; R 1 u + (; ) ^F (; v + (); ) di where the linear part L : Y! ^Y is bounded and given by 1 C A ; and 1 C C A ; L(b + ; b ; v + ; v ; ) = v + s +(; )b + v u (; )b R R s +(; )D f(h(); ) d + R 1 u +(; )D f(h(); ) d u (; )D f(h(); ) d + R 1 s (; )D f(h(); ) d b + b R 1 u +(; )D f(h(); ) d + R 1 s (; )D f(h(); ) d h'; s +(; )b + R 1 u +(; )D f(h(); ) di 1 C A : Note that the last two components of L do not depend on (v + ; v ). The linear operator : R( s +(; )) R( u (; ))! X ; (b + ; b ) = b + b ; is a Fredholm operator with index zero. Its null space and range are given by N( ) = spanf( h(); _ h())g _ R( s +(; )) R( u (; )); R( ) = fv 2 X ; h (); vi = g: 18

20 On the other hand, h'; s +(; ) _ h()i = h'; _ h()i = 1; and Z 1 1 = h (); D f(h(); )i d D (); Z 1 u +(; )D f(h(); ) d + Z 1 E s (; )D f(h(); ) d : Hence, as a consequence of Hypothesis (H3), the operator L is continuously invertible. Due to the estimate (4.3), we can apply Lemma 4.1 for any small xed to the map G with y = (; ; ; ). Hence, we obtain the existence and uniqueness statements in (ii) and (iii) of Theorem 1. It is straightforward to show that h (t) is homoclinic to the hyperbolic equilibrium p ( ). The estimate given in (ii) follows from Lemma 4.1 provided we can prove that where G (; ; ; ; ) = jg (; ; ; )j ^Y C sup j(id Q )h(t)j ; (4.4) t2r R t s +(t; )(id Q )f(h(); ) d + R t 1 u +(t; )(id Q )f(h(); ) d R t u (t; )(id Q )f(h(); ) d + R t 1 s (t; )(id Q )f(h); ) d R 1 u +(; )(id Q )f(h(); ) d + R 1 s (; )(id Q )f(h(); ) d h'; R 1 u + (; )(id Q )f(h(); ) di In order to prove (4.4), it suces to show that Z t s +(t; )(id Q )f(h(); ) d C sup 2R 1 C A : j(id Q )h()j (4.5) for some constant C independently of t, and similar estimates for the other integrals. We use the fact that h(t) satises _ h = Ah + f(h; ) for t 2 R, that is, Therefore, using (Q)(i), we have Z t = = s +(t; )(id Q )f(h(); ) d Z t Z t h 2 C 1 (R; X) \ C (R; D(A)): (4.6) s +(t; )(id Q )( d h() Ah()) d d d d s +(t; ) (id Q )h() s +(t; )A(id Q )h() d 19

21 = = + s +(t; t)(id Q )h(t) Z t s +(t; )(A + D u f(h(); ))(id Q )h() s +(t; )A(id Q )h() d + s +(t; t)(id Q )h(t) Z t s +(t; )D u f(h(); )(id Q )h() d + s +(t; t)(id Q )h(t): Note that integration by parts and taking the derivative d d s +(t; ) is allowed on account of (4.6). It is now straightforward to obtain the aforementioned estimate (4.5). The other estimates are obtained in an analogous fashion, and we omit the details. This proves the claim (4.4). Finally, we show the homoclinic orbits h (t) of (2.4) are non-degenerate. Lemma 4.3 The only bounded solution, up to constant multiples, of the variational equation _v = (A + Q D u f(h (t); ))v about h (t) is given by h _ (t). In other words, the solutions h (t) are non-degenerate. Proof. On account of Theorem 5, the variational equation _v = Av + Q D u f(h (t); )v has an exponential dichotomy on R + and R with solution operators s +;(t; ) and u +;(; t) for t, and s ;(; t) and u ;(t; ) for t. On account of Hypothesis (C), and Lemmata 3.1 and 4.2, s +;(; ) and u ;(; ) are close to s +(; ) and u (; ), respectively, in the L(X )-norm. Therefore, h _ (t) is the only bounded solution, up to constant multiples, of equation (2.4). It is a consequence of the proof of Lemma 4.3 that h _ () 2 C (R; X ). Indeed, h _ () 2 R( s +;(; )), and therefore h _ () 2 X. Furthermore, h _ (t) = s +;(t; ) h _ () for t > is continuous in t as a function into X by Theorem 5. Since the choice of t = is arbitrary, we see that in fact h _ () 2 C (R; X ). 5 The Truncated Boundary-Value Problem In this section, we prove Theorem 2. Again, C denotes various dierent constants independent of T and T +. 2

22 5.1 The Nonlinear Equation We exploit the transformation u(t) = h (t) + v(t) and = +. The function v(t) then satises the equation _v = (A + D u f(h(t); ))v + F (t; v; ) (5.1) = (A + D u f(h(t); ))v + D f(h (t); ) + ^F (t; v; ); where ^F (t; v; ) = Q f(h (t) + v; + ) f(h (t); ) D u f(h(t); )v The derivative D (v;) ^F (t; v; ) is given by (id Q )D u f(h(t); )v D f(h (t); ): D (v;) ^F (t; v; ) = h Q (D u f(h (t) + v; + ) D u f(h(t); )) (id Q )D u f(h(t); ); Q (D f(h (t) + v; + ) D f(h (t); )) (id Q )D f(h (t); ) Due to Theorem 1, Hypothesis (C), and Lemma 4.2, we obtain the estimate kd (v;) ^F (t; v; )k L(X R;X) C(jvj + jj) + g(); (5.2) uniformly in for (v; ) in a ball centered at zero of suciently small radius in X R. Here, the function g()! as!. Let We dene the maps by a = (a + ; a ) 2 X a := R(P + ) R(P ); (5.3) b = (b + ; b ) 2 X b := R( s +(; )) R( u (; )): I +;T; : X a X b C ([; T + ]; X ) R! C ([; T + ]; X ); I ;T; : X a X b C ([T ; ]; X ) R! C ([T ; ]; X ); I +;T; (a; b; v + ; )(t) = u +(t; T + )a + + s +(t; )b + (5.4) + Z t T + u +(t; )F (; v + (); ) d + Z t s +(t; )F (; v + (); ) d; and the analogous expression for I ;T; (a; b; v ; ). Note that both maps are smooth. Any bounded solution of (5.1) satises the integral equation = v + (t) I +;T; (a; b; v + ; )(t); (5.5) = v (t) I ;T; (a; b; v ; )(t); 21 i :

23 together with the equation v + () = v () for some (a; b). Here, t 2 [; T + ] in the rst, and t 2 [T ; ] in the second equation in (5.5). In addition, we have to solve the phase and boundary conditions R (h (T + ) + v + (T + ); h (T ) + v (T ); + ) = ; (5.6) J T; (h + V (v + ; v ); + ) = ; where the linear, bounded operator V : C ([; T + ]; X ) C ([T ; ]; X )! C ([T ; T + ]; X ) is dened by V (v + ; v )(t) = For v 2 X, we expand the boundary conditions 8 < : v +(t) + v () v + () t > ; v (t) t : (5.7) R (h (T + ) + v + ; h (T ) + v ; + ) = R (h (T + ); h (T ); ) (5.8) +D u+;u ;R (h (T + ); h (T ); )(v + ; v ; ) + ^R (h (T + ); h (T ); v + ; v ; ); with kd (v+;v ;) ^R (h (T + ); h (T ); v + ; v ; )k C(jv + j + jv j + jj); (5.9) for (v + ; v ; ) in a ball with small radius centered at zero in X X R, independently of. Similarly, for v 2 C (T; X ), we have J T; (h + v; + ) = J T; (h ; ) (5.1) +D v J T; (h ; )v + D J T; (h ; ) + ^JT; (h ; v; ); with kd (v;) ^JT; (h ; v; )k C(jvj + jj); (5.11) for (v; ) in a ball with small radius centered at zero in C (T; X ) R, independently of. We consider the nonlinear equation G T; : Y! ^Y ; GT; (a; b; v + ; v ; ) = (5.12) with Y = X a X b C ([; T + ]; X ) C ([T ; ]; X ) R ^Y = C ([; T + ]; X ) C ([T ; ]; X ) X X R; 22

24 dened by the right-hand side of (5.5), the continuity equation = I +;T; (a; b; v + ; )() I ;T; (a; b; v ; )() the rst two equations in (5.6), and the equation = J T; (h + V (I +;T; (a; b; v + ; ); I ;T; (a; b; v ; )); + ): It is a consequence of the above discussion that G is well dened and smooth. Furthermore, due to the estimates (5.2), (5.9) and (5.11), we can solve equation (5.12) in a ball centered at the origin with small radius uniformly for any suciently small provided the linearized operator at (a; b; v + ; v ; ) = is invertible uniformly in T and. The arguments are analogous to those presented in the last section, whence we omit them. Note that the error estimate in Theorem 2 follows from G T; () = ; ; ; R (h (T + ); h (T ); ); J T; (h ; ) and Lemma 4.1. Indeed, replacing h () by h ( + T; ) for some small T;, we can achieve that J T; (h ; ) =. 5.2 The Linearized Boundary-Value Problem It remains to show that the operator L T; = DG T; () is invertible as a map from Y to ^Y. Let ^I +;T; (a; b; )(t) = u (t; T + +)a + + s (t; )b + + Z t Z t (5.13) + u +(t; )D f(h (); ) d + s +(t; )D f(h (); ) d T + ; ^I ;T; (a; b; )(t) = s (t; T )a + u (t; )b (5.14) Z t Z t + s (t; )D f(h (); ) d + u (t; )D f(h (); ) d ; T for t 2 [; T + ] and t 2 [T ; ], respectively. The linear operators ^I are bounded from X a X b R into C ([; T + ]; X ) and C ([T ; ]; X ), respectively. We then have L T; (a; b; v + ; v ; ) = (5.15) B v + ^I+;T; (a; b; ) v ^I ;T; (a; b; ) ^I +;T; (a; b; )() ^I ;T; (a; b; )() DR (h (T + ); h (T ); )^I+;T; (a; b; )(T + ); ^I ;T; (a; b; )(T ); D v J T; (h ; )V (^I +;T; (a; b; ); ^I ;T; (a; b; )) + D J T; (h + v; ) 23 1 C C A :

25 We have to show that the equation L T; (a; b; v + ; v ; ) = (g + ; g ; c; r; j) (5.16) has a unique solution (a; b; v + ; v ; ) 2 Y for any (g + ; g ; c; r; j) 2 ^Y, and j(a; b; v + ; v ; )j Y Cj(g + ; g ; c; r; j)j ^Y for some positive constant C independent of and T. Inspecting the denition (5.15) of L T;, it is clear that we can solve the rst two components in (5.16) for (v + ; v ) such that (v + ; v ) = W 1 (a; b; ; g + ; g ) with kw 1 k C. Next, consider the boundary condition r = DR (h (T + ); h (T ); )(w + ; w ; ); (5.17) with w + = u +(T + ; T + )a + + s +(T + ; )b + + w = s (T ; T )a + u (T ; )b + The key is that Z Z T + s +(T + ; )D f(h (); ) d; u (T ; )D f(h (); ) d: T D u+;u R (h (T + ); h (T ); )( u (T + +; T + )j R(P+); s (T ; T )j R(P ) ) : R(P + ) R(P )! X is invertible uniformly in due to Hypothesis (T1)(ii). Indeed, the projections u +(T + ; T + ) and P + as well as s (T ; T ) and P are close to each other for all jt j, T + suciently large and small enough due to Hypothesis (C), and Lemmata 3.1 and 4.2. Therefore, we can solve (5.17) for a = (a + ; a ) and obtain a = W 2 (b; ; r) with kw 2 k C independently of T, T + and. Actually, we obtain the better estimate jw 2 (b; ; r)j C(e T+ jb + j + e T jb j + jj + jrj ); since j s +(T + ; )b + j Ce T+ jb + j ; and the analogous estimate for u (T ; ) by Theorem 5. In the next step, we apply these estimates to the operator V (^I+;T; (a; b; ); ^I ;T; (a; b; )) appearing in the phase condition; see also (5.7) for its denition. Using (5.13), (5.14) and the estimates for a, we obtain the expansion V (^I +;T; ; ^I ;T; )(a; b; )(t) = 8 < : s +(t; )b + + b b + + W 3 (b; ; r)(t) t > ; u (t; )b + W 3 (b; ; r)(t) t ; 24

26 with jw 3 (b; ; r)(t)j C(e T+ jb + j + e T jb j + jj + jrj ): According to the results in Section 4.2, we may write (b + ; b ) = (^b + ; ^b ) + ( h(); _ h()); _ (^b + ; ^b ) 2 ^X b with ^X b spanf( h(); _ h())g _ = Xb. We obtain V (^I +;T; ; ^I ;T; )(a; b; )(t) = h(t) _ + W 4 (b; ; r)(t); with jw 4 (b; ; r)(t)j C(e jt j jj + j^b + j + j^b j + jj + jrj ); and jt j := min(jt j; T + ). The phase condition and continuity equation are then given by j = D v J T; (h ; ) _ h + W 5 (b; ; r); c = u +(; T + )W 2;+ (b + ; ; r + ) s (; T )W 2; (b ; ; r ) + ^b + ^b (5.18) Z T+ Z u +(; )D f(h (); ) d + s (; )D f(h (); ) d ; T with jw 5 (b; ; r)j C(e T+ jb + j + e T jb j + jj + jrj ): Note that we have the estimates due to Theorem 5. Moreover, j u +(; T + )W 2;+ (b + ; ; r + ) s (; T )W 2; (b ; ; r )j C(e T+ + e T )(jb + j + jb j + jj + jrj ) Z T+ T h (t); D f(h (t); )i dt is bounded away from zero due to Hypothesis (H3), Theorem 1 and the fact that (t) converges to zero exponentially. Therefore, by the same arguments as in Section 4.2 using Theorem 1 and Hypothesis (T1)(i), we can solve (5.18) for (^b; ; ). 5.3 Proof of Theorem 4 The proof is a consequence of the proof of Theorem 2. In fact, we only need to consider functions v + and variables a + and b +. It is straightforward to see that b + can be used to solve the boundary condition (id S)u() = due to the transversality condition (R)(ii). We omit the details. 25

27 6 The Finite-Dimensional Boundary-Value Problem In this section, we prove Theorem 3. We embed the boundary-value problem on R(Q ) into a larger one dened on X, and then apply Theorem 2. Any element u in X can be written according to Using this decomposition, we have u = q + w; (q; w) 2 R(Q ) N(Q ): A + Q D u f(p (); ) (A + Q D u f(p (); ))j R(Q) Q D u f(p (); )j N (Q) Aj N (Q) The spectral projections of A, A + Q D u f(p (); ) and (A + Q D u f(p (); ))j R(Q) are denoted by ^P, P ; () and Q ; (), respectively; see Hypotheses (A1) and (H1). On account of Hypothesis (Q), we then have for some bounded operators D ; (). P ; () Q ;() D ; () (id Q ) ^P The equation _u = Au + Q f(u; ) is equivalent to We include the phase and boundary conditions 1 A ; 1 A : _q = Aq + Q f(q + w; ); _w = Aw: (6.1) Z T+ ~J((q; w); ) = h h _ (t); (q(t) + w(t) h (t)i X dt; T ~R + ((q; w)(t + ); ) = P +; ( )P +; ()(q(t + ) + w(t + ) p ()) (6.2) ~R ((q; w)(t ); ) = P ; ( )P ; ()(q(t ) + w(t ) p ()): We rst prove that (6.1{6.2) has a unique solution. Using Remark 2.1, it is straightforward to show that (6.2) satises Hypothesis (T1). For instance, for =, D u R+ ~ (p ( ); ) = P ; ( ) R(P;( )) R(P;( )) which is clearly invertible as an operator into R(P +; ( )). Therefore, this operator remains invertible for close to with uniform inverse. The same argument applies to the derivative of the second boundary condition. Hence, Theorem 2 applies, and (6.1{6.2) has a unique solution. To nish the argument, we observe that any solution (q; w) of (6.1{6.2) has necessarily w =. Indeed, w has to satisfy _w = Aw; ^P+ w(t + ); ^P w(t ) = : 26

28 Since Aj N (Q) is hyperbolic, w = is the only solution. With w =, it is easy to see that (6.1{6.2) and (2.6{2.8) coincide. Hence, (q; w) = (q; ) satises (6.1{6.2) if, and only if, q is a solution of (2.6{2.8). Finally, we have jr +; (h(t + ); )j Cjh(T + ); )j 2 Ce 2s T + ; and the analogous estimate for R +; (h(t + ); ). This completes the proof of Theorem 3. 7 Semilinear Elliptic Equations Here, we show that elliptic equations on innite cylinders are included in the abstract set-up of the earlier sections. We refer to [26] for more details. Furthermore, we comment on the satisfaction of the hypotheses of Theorems 1 and 2, and discuss particular discretizations. Let Y be a Hilbert space and L : D(L) Y! Y a densely dened, positive denite, self-adjoint operator with compact resolvent. In most application, we have Y = L 2 () for some bounded domain and L = on together with Dirichlet boundary conditions, say, so that D(L) = H 2 () \ H 1 (). We denote the fractional power spaces associated with L by Y. In particular, Y 1 = D(L). Suppose that g : Y 1+ 2 Y 2! Y is a nonlinearity of class C 2 for some 2 [; 1) and >. Consider the abstract elliptic equation for u 2 Y. We reformulate (7.1) as the rst-order equation with v = (u; u x ) and G(v) = (; g(v)). Here, u xx Lu = g(u; u x ); x 2 R (7.1) A id L d v = Av + G(v) (7.2) dx 1 A : Y 1 Y 1 2! Y 1 2 Y: In particular, Hypothesis (A1) is met; in fact, the projections ^P are given by ^P = 1 id L 1 2 L 1 2 id 1 A : Y 1 2 Y! Y 1 2 Y: The fractional power spaces are X = Y 1+ 2 Y 2. The mapping G : X X! X is C 2 since g is. It is also clear that A has compact resolvent whenever L has. 27

29 Therefore, Hypotheses (A1), (A2) and (C) are met. We refer to [8, Satz 5] for conditions guaranteeing that Hypothesis (A3) is met. Given a particular solitary-wave solution of such an elliptic system, hyperbolicity of equilibria (H1) and transverse unfolding (H3) are generic properties, at least if we allow for nonlinearities of the form g(y; u; u x ; r y u; ). In order to apply our results to concrete problems, we have to choose a discretization in the cross-section, corresponding to the projectors Q, and boundary conditions at x = T and x = T +. For elliptic equations (7.1), it is convenient to choose Q with 2 f1=k; k 2 Ng as the orthogonal Galerkin projections onto the rst m eigenfunctions of L. Condition (Q) is then an immediate consequence of the completeness of the orthonormal system of eigenfunctions. The choice of boundary conditions R turns out to be less evident in general, as the projectors P +; and P ; might be hard to compute. We emphasize here that, in general, simple Dirichlet boundary conditions u(t ) = p or Neumann boundary conditions v(t ) = will not work. Even for systems of equations on the line with no cross-section, that is Y = R 2k, the dimensions of stable and unstable subspaces at the equilibrium may not coincide: dim R(P + ) 6= k. Then Dirichlet as well as Neumann boundary conditions yield ill-posed problems. The only generic choice then seems to be given through periodic boundary conditions or the actual computation of P +. However, there are important cases where Dirichlet and Neumann conditions work. Examples are reversible systems or equations of variational type, which we now discuss in more detail. If g = g(u), then the system is reversible. Reversibility acts through S(u; v) = (u; v). The condition (id S)u() in (2.1) reduces to v =, in other words, Neumann boundary conditions at x =. The hyperbolicity assumption (H1) is then equivalent to linear stability of the equilibrium u(x; y) = p(y) for the parabolic equation u t = u xx Lu g(u) on the cylinder. Due to the second-order structure, eigenfunctions of the linearization of (7.1) at the equilibrium are of the form (u k ; p k u k ) where k and u k are eigenvalues and eigenfunctions, respectively, of L + Dg(p). By hyperbolicity, k >. We claim that we can choose Dirichlet or Neumann conditions at x = T + as well. Indeed, the stable subspace R(P ) is spanned by (u k ; p k u k ) and the spaces f(u; v); u = g or f(u; v); v = g are closed complements of this subspace. We summarize this result in the following proposition. Proposition 1 Assume that (H1), (H2) and (H4) are met. g = g(u). Dirichlet and Neumann boundary condition then satisfy (T2). Furthermore, suppose that These arguments can be slightly generalized to elliptic equations with variational structure u xx = Lu + cu x + rf (u); 28