NUMERICAL COMPUTATION OF SOLITARY

NUMERICAL COMPUTATION OF SOLITARY WAVES IN INFINITE CYLINDRICAL DOMAINS GABRIEL J. LORD, DANIELA PETERHOFy , BJO RN SANDSTEDEz , AND ARND SCHEELx

Abstract. The numerical computation of solitary waves to semilinear elliptic equations in in nite cylindrical domains is investigated. Rather than solving on the in nite cylinder, the equation is approximated by a boundary-value problem on a nite cylinder. Convergence and stability results for this approach are given. It is also shown that Galerkin approximations can be used to compute solitary waves of the elliptic problem on the nite cylinder. In addition, it is demonstrated that the aforementioned procedures simplify in cases where the elliptic equation admits an additional reversibility structure. Finally, the theoretical predictions are compared with numerical computations. In particular, post buckling of an in nitely long cylindrical shell under axial compression is considered; it is shown numerically that, for a xed spatial truncation, the error in the truncation scales with the length of the cylinder as predicted theoretically. Key words. Solitary wave, boundary-value problem, elliptic equation AMS subject classi cations. 65N12, 35B30, 35J55, 65N30

1. Introduction. The numerical computation of solitary-wave solutions to el-

liptic systems (1.1)

uxx + y u + g(y; u; ux; ry u) = 0;

(x; y) 2 R

in in nite cylinders R is investigated. Here, u 2 Rm , and is an open and bounded subset of Rn with Lipschitz boundary. Appropriate boundary conditions (1.2)

R((u; ux; ry u)jR@ ) = 0

on R @ should be added. Solitary waves are solutions h(x; y) that satisfy 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 (1.3)

ut = uxx + y u + g(y; u; ux; ry u);

(x; y) 2 R :

These applications include problems in structural mechanics such as shells and struts, chemical kinetics, combustion, and nerve impulses; see, for instance, [34] and the comprehensive bibliography there. Analytically, the existence of solitary-wave solutions that have a non-trivial structure in the cross-section is still a largely open problem. Existence has been proved in many cases for small solutions using center-manifold theory [20, 26]. In special cases, it may be possible to exploit maximum principles CISE, National Physical Laboratory, Queens Road, Teddington TW11 0LW, UK. Supported by the EPSRC, UK. y Weierstra-Institut f ur Angewandte Analysis und Stochastik, Mohrenstrae 39, 10117 Berlin, Germany. Supported by the Deutsche Forschungsgemeinschaft (DFG) under grant Schn426/5-1. z Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210, USA. x Institut f ur Mathematik I, Freie Universitat Berlin, Arnimallee 2-6, 14195 Berlin, Germany. 1

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G. LORD, D. PETERHOF, B. SANDSTEDE, AND A. SCHEEL

[1, 2, 18] and variational structure [28, 29]. Another approach uses topological methods [11, 14] to establish the existence of front solutions. Suppose that h(x; y) is a solitary wave that satis es (1.1{1.2). To calculate h numerically, the problem on the in nite cylinder R has to be approximated by an appropriate system (1.4)

uxx + y u + g(y; u; ux; ry u) = 0; R((u; ux; ry u)j[T? ;T+ ]@ ) = 0

(x; y) 2 (T? ; T+ ) ;

that is posed on a nite cylinder. Here, we have to specify additional conditions (1.5)

R? ((u; ux; ry u)jfT? g ) = 0; R+ ((u; ux; ry u)jfT+g ) = 0

at the boundaries that are induced by the truncation of the cylinder axis. The issue is then to determine whether (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. In this article, we give sucient conditions on the equation and the boundary conditions (1.5) such that the aforementioned approach works. Boundary conditions that satisfy these requirements 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 hT of (1.4{1.5) and the solitary wave h can then be estimated by

jh ? hT j C (jR? (hjfT? g )j + jR+ (hjfT+ g )j) in appropriate norms, where the positive constant C does not depend on T? and T+ . Note that the right-hand side converges to zero exponentially as jT j ! 1. To prove this result, we interpret the variable x as time and write (1.1) as a rst-order system

u

v (1.6) ?y u ? g(y; u; v; ry u) : Here, for each xed x 2 R, (u; v)(x) is a function of y 2 that is contained in some function space that depends on the boundary conditions on @ . A solitary wave of x vx =

(1.1) corresponds to a homoclinic or heteroclinic solution of (1.6) that connects the equilibria p? (y) and p+ (y), i.e., we have

(1.7)

lim (h(x); hx (x)) = (p ; 0)

x!1

in the underlying function space. We then replace (1.7) by a condition of the form (1.8)

R? ((u; v)(T? )) = 0;

R+ ((u; v)(T+ )) = 0

and investigate the resulting truncated boundary-value problem. The key for solving this boundary-value problem are exponential dichotomies for the linearization u ux 0 id (1.9) vx = ?y ? Du g ? Dry u g ry Dux g v

NUMERICAL COMPUTATION OF SOLITARY WAVES IN INFINITE CYLINDERS

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of (1.6) about the solitary wave (h(x); hx (x)). Here, derivatives of g are evaluated at (y; h; hx; ry 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)(x0 ) in the stable space E s (x0 ) exist for x > x0 and decay exponentially for x ! 1. In contrast, solutions (u; v)(x) associated with initial values (u; v)(x0 ) in the unstable space E u (x0 ) satisfy (1.9) in backward xdirection x < x0 and decay exponentially for decreasing x. In the context of elliptic equations, the stable and unstable spaces are both in nite-dimensional. The existence of exponential dichotomies for ordinary, parabolic or functional dierential equations is well known. For elliptic equations, existence has recently been proved in [27] using a novel functional-analytic approach. The results in this latter article 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. First, we concentrate on the elliptic formulation. Consider, for instance, equation (1.1) with Dirichlet boundary conditions, i.e. uxx + y u + g(y; u; ux; ry u) = 0; (x; y) 2 R ; ujR@ = 0 and assume that the solitary wave converges to zero as jxj tends to in nity. We may then want to take Dirichlet boundary conditions for the arti cial conditions (1.5) which results in the truncated problem uxx + y u + g(y; u; ux; ry u) = 0; (x; y) 2 (T? ; T+ ) ; uj@ ((T?;T+ ) ) = 0: 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. Second, we could take a dynamical-systems point of view and consider the rst-order system (1.6{1.8). We then discretize only in the crosssection and obtain a large system of ODEs

u

v ?y u ? g(y; u; v; ry u) ; 0 = R? ((u; v)(T? )); 0 = R+ ((u; v)(T+ )) de ned on the range R(Qn ) where the Galerkin projection Qn projects the function x vx = Qn

space in onto a nite-dimensional subspace. ODE codes such as Homcont, see [8, 10], can now be used to solve the resulting boundary-value problem. Often, elliptic equations have an additional re ection symmetry. For instance, consider the fourth-order equation uxxxx + 2y u + g(y; u; uxx; y u) = 0; (x; y) 2 R ; (1.10) which is included in our general set-up. The Z2-symmetry u(x; y) 7! u(?x; y) leaves (1.10) invariant. This symmetry manifests itself as a time-reversibility S : (u; v1 ; v2 ; v3 ) 7! (u; ?v1 ; v2 ; ?v3 ) for the associated dynamical system (u; v1 ; v2 ; v3 )x = (v1 ; v2 ; v3 ; ?2y u ? g(y; u; v2 ; y u)):

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It is shown that the algorithm given above can be adapted to this situation. We also emphasize that our results apply to parabolic equations (1.11) ut = u + g(u; ru); x 2

with Rn bounded and open. Here, we are interested in the computation of homoclinic or heteroclinic solutions h(t; x) that satisfy limt!1 h(t; x) = p (x). For ordinary dierential equations, well-posedness of the truncated problem has been investigated by Beyn [4] and Doedel & Friedman [13] for very general boundary conditions. In addition, error estimates have been derived in these articles. Hagstrom & Keller [16, 17] considered elliptic problems of the form (1.1) assuming that (1.9) has an exponential dichotomy. They investigated the so-called asymptotic boundary conditions that select precisely those solutions that converge to p (y) as x ! 1. In particular, the solution of the truncated problem coincides with the true wave h on the in nite cylinder. The actual calculation of the asymptotic boundary conditions, however, involves again certain approximations which were not investigated in [16, 17]. As a concrete application, we consider the post buckling of an in nitely long cylindrical shell under axial compression as modeled by the von Karman{Donnell equations. In [23, 24, 25], 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 [27] in x3. The theorems on Galerkin approximations and the truncated boundary-value problem are proved in x4{6. We show in x7 that our results apply to semilinear elliptic equations. Numerical simulations and a comparison of the numerical and theoretical error are presented in x8, while the application to the von Karman{Donnell equation is given in x9. Finally, we summarize our results in x10.

2. Main Results. 2.1. The Setting. Assume that A is a closed operator de ned on a re exive 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 B maps D(A) into itself and AB = BA on D(A). Hypothesis (A1). Suppose that there is a constant C such that

k(A ? i)?1 kL(X ) 1 +Cjj

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 > 0 such that Re < ? for any 2 spec(AP^? ) and Re > for any 2 spec(A(id ?P^? )). Sucient conditions for the existence of the projection P^? , which is sometimes referred to as the Calderon projector [7], have been given in [6, 15]. We also refer to the explicit construction of this projection for semilinear elliptic equations in [27, x6]. De ne P^+ = id ?P^? and A? = ?P^? A, A+ = P^+ A, and let X? = R(P^? ) and X+ = R(P^+ ). Here, R(L) and N(L) denote the range and null space, respectively,


5

of an operator L. By Hypothesis (A1), the operators A? and A+ are sectorial with their spectrum contained in the right half-plane. Therefore, for 0, we can de ne the interpolation spaces X+ = D(A+ ) and X? = D(A? ), see [19]. Finally, we set X = X+ X? . We denote the norm in X by jj and the operator norm in L(X ) by k k . The projection P^? is then in L(X ) for any < 1. 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 for some xed 2 [0; 1) and for f 2 C 2 (X R; X ). We say that u(t) is a solution of (2.1) on the interval [0; T ) if u 2 C 1 ((0; T ); X ) \ C 0((0; T ); D(A)) \ C 0 ([0; T ); X ) and u(t) satis es (2.1) in X for t 2 (0; T ). We assume the existence of a hyperbolic equilibrium and a homoclinic orbit at = 0. Hypothesis (H1). Equation (2.1) has a hyperbolic equilibrium p0 2 D(A) for = 0. In particular, Hypothesis (A1) is met with A replaced by A + Du f (p0 ; 0). Hypothesis (H2). Let h(t) 2 C 1(R; X ) \ C 0 (R; X 1 ) be a homoclinic solution of (2.1) for = 0 with h(t) ! p0 as jtj ! 1. We assume that h(t) is nondegenerate, i.e. ddt h(t) is the only bounded solution, up to constant multiples, of the variational

(2.1)

equation

v_ = Av + Du f (h(t); 0)v:

(2.2)

Next, we introduce the adjoint variational equation

v_ = ?(A + Du f (h(t); 0) )v about the homoclinic solution h(t). The operator A is again closed and densely de ned in X . To describe the asymptotic behavior of solutions of (2.2) and (2.3), we (2.3)

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(0) = 0 is the trivial solution v(t) = 0. 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) doesZ 1 not vanish. Hypothesis (H3). M := h (t); D f (h(t); 0)i dt 6= 0. ?1

2.2. The Galerkin Approximation. First, we show the persistence of the homoclinic orbit h(t) under nite-dimensional Galerkin approximations of (2.1). We may think of a Galerkin approximation as a family of projections denoted by Q 2 L(X ) for > 0. Here, Q0 = id, while Q typically has nite-dimensional range for > 0 and approximates the identity in a weak sense. Hypothesis (Q).

(i) A commutes with Q . (ii) The norms kQ kL(X ) C are bounded uniformly in . (iii) For any u 2 X , we have jQ u ? uj0 ! 0 as ! 0.

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It is a consequence of Hypothesis (Q)(i) that A Q = Q A . Therefore, we have Q 2 L(X ) and kQ kL(X ) C independently of > 0. Furthermore, jQ u ? uj ! 0 as ! 0 for any u 2 X . In order to obtain uniform convergence of the Galerkin approximation, we assume compactness of the nonlinearity f . Hypothesis (K). If Q 6= id for some > 0, we assume that f (; 0) : X ! X is a compact map for = 0. The next theorem shows the persistence of the equilibrium and the homoclinic orbit for the Galerkin approximation (2.4)

u_ = Au + Q f (u; );

(u; ) 2 X R

of (2.1). We emphasize that the subspaces R(Q ) and N(Q ) are both invariant under equation (2.4). For initial data u0 2 (id ?Q)X , equation (2.4) reduces to the linear equation u_ = Au, which has no bounded solution on R except u = 0. We also remark that the norms on Q X and Q X are equivalent if the range of Q is nite-dimensional; the equivalence constants, however, tend to in nity as ! 0. Therefore, estimates which are uniform with respect to can only be expected in the X -norm. Theorem 2.1. Assume that Hypotheses (A1){(A3), (H1){(H3), (K) and (Q) are satis ed. There are then positive numbers 0 , 0 , and C such that the following is true for any 0 < 0 and jj < 0 . (i) Equation (2.4) has a hyperbolic equilibrium p () 2 R(Q ) with p0 (0) = p0 and

jp () ? p0 j C (j(id ?Q)p0 j + jj): (ii) For every , there exists a such that (2.4) has a nondegenerate homoclinic orbit h (t) 2 Q X with h (t) ! p ( ) as jtj ! 1, and

j j + sup jh (t) ? h(t)j C sup j(id ?Q )h(t)j : t2R

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 < 0 g. We denote the spectral projections associated with A + Du f (p0; 0) and A + Q Du f (p (); ) in X by P and P; (), respectively. In particular, P? and P?; () project onto the stable eigenspaces corresponding to eigenvalues of A + Du f (p0 ; 0) and A + Q Du f (p (); ), respectively, with negative real part. 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 in nite interval R to a nite interval [T? ; T+] for some T? < 0 < T+ and imposing boundary conditions at the end points t = T? and t = T+ . The truncated boundary-value problem is given by 0 u_ ? Au ? Q f (u; ) 1 @ R (u(T+); u(T?); ) A = 0 (2.5) JT; (u; ) for t 2 T = (T? ; T+). Here, JT; denotes a phase condition and R encodes the boundary conditions; note that any translate h( + ) of the homoclinic orbit h() is

7


again homoclinic. The phase condition is needed to pick one particular translate and thus to make the solution unique. We refer to x2.4, and in particular to (2.7) and (2.8), for examples of phase and boundary conditions that are often used in practice. We assume that the following hypotheses are met.

Hypothesis (T1). (i) JT; : C 0 (T; X ) R ! R is of class C 2 , and JT; (h ; ) ! 0 as jTj ! 1. Furthermore, there is a constant d0 > 0 independent of T?, T+ and such that Du JT; (h ; )h_ d0 > 0 for all jTj suciently large. Finally, Du JT; (u; ) and D2u JT; (u; ) are bounded in a ball in C 0 (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 D2 R are bounded in a small ball centered at (p ( ); p ( ); ) in X X R uniformly in . Fur-

thermore,

D(u+ ;u? ) R (p ( ); p ( ); )jR(P+; ( ))R(P?; ( )) is invertible, and the inverse is bounded uniformly in . Note that h_ () and h_ () are contained in C 0 (T; X ). Therefore, the condition on J in (T1)(i) makes sense. Remark 2.2. The boundary conditions are often separated, i.e. given by

R (u+ ; u?; ) = (R+; (u+ ; ); R?; (u? ; )) 2 R(P+; ( )) R(P?; ( )) = X : If the operators Du R; (p ( ); )jR(P; ( )) are invertible, and the inverses are bounded uniformly in , then the invertibility condition in Hypothesis (T1)(ii) is also satis ed.

Notation. Throughout this article, C denotes various dierent constants all independent of T? and T+. We have the following theorem. Theorem 2.3. Assume that (A1){(A3), (H1){(H3), (K), (Q) and (T1) are met. There exist positive numbers 0 , and C such that the following is true for all suciently large intervals T . For any 2 [0; 0 ), the boundary-value problem (2.5) has a unique solution (h (t); ) in the tube

f(u; ) 2 C 0 ([T? ; T+]; X ) R; jj + sup ju(t) ? h(t)j g; t2[T? ;T+ ]

and

j ? j + sup jh (t) ? h (t + T; )j C jR (h (T+ ); h (T? ); )j t2[T? ;T+ ]

for an appropriate small number T; . Combining Theorems 2.1 and 2.3, and exploiting Hypothesis (T1), we obtain the following corollary. Corollary 2.4. Under the assumptions of Theorem 2.3, we have the estimate

j j + sup jh (t) ? h(t)j j C jR (h(T+ ); h(T? ); 0)j + sup j(id ?Q )h(t)j t2[T? ;T+ ]

t2R

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).

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The error estimate can be made more explicit. Corollary 2.5. Under the assumptions of Theorem 2.3, we have the estimate j j + sup jh (t) ? h(t)j C (es T+ + eu T? + sup j(id ?Q )h(t)j ); t2R

t2[T? ;T+ ]

where the constants s < 0 and u > 0 are chosen such that 2= spec(A + Du f (p0 ; 0)) for any 2 C with s Re u . We point out that the case Q = id for all is included in the analysis. It corresponds to truncating (2.1) directly without going to a nite-dimensional approximation. A more analytical consequence of Theorem 2.3 is the existence of periodic solutions with large period near the homoclinic orbit h. Corollary 2.6. Assume that (A1){(A3) and (H1){(H3) are met. There is then a constant > 0 such that (2.1) has a periodic orbit (u ; ) with minimal period 2 for any > . Furthermore, j j + sup ju (t) ? h(t)j C (es + e?u ); t2[?; ] where the constants s and u are as in Corollary 2.5. Proof. Consider the phase condition JT (u) := h'; u(0)i where ' 2 (X ) is chosen such that h'; h_ (0)i = 1. The boundary condition is R(u+ ; u?; ) = u+ ? u? . Since R(P+ ) R(P? ) = X ,Hypothesis (T1) is satis ed, and we can apply Theorem 2.3 with Q = id for all .

This corollary has been proved for ordinary dierential equations in [3] and [22]. The proof given in [22] 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 space R(Q ), i.e. (2.6) q_ = Aq + Q f (q; ); (q; ) 2 R(Q ) R: Note that R(Q ) is typically nite-dimensional though we do not assume that this is the case. If X is a Hilbert space, the phase condition may be chosen according to (2.7)

JT; (q; ) =

Z T+ T?

hh_ (t); q(t) ? h (t)iX dt:

For the boundary conditions, we may take, for instance, the projection boundary conditions which are de ned by (2.8) R+; (q(T+ ); ) = Q+; ()(q(T+ ) ? p ()); 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 Du f (p (); ))jR(Q ) . We have the following result for the nite-dimensional boundary-value problem on R(Q ) that we described above. Theorem 2.7. Assume that (A1){(A3), (H1){(H3) and (K) are met where X is a Hilbert space. There exist positive numbers 0 , , and C such that the following is true for all suciently large intervals T . For any 2 [0; 0 ), the boundary-value problem (2.6{2.8) on R(Q ) has a unique solution (h (t); ) in the tube f(q; ) 2 C 0 ([T? ; T+ ]; R(Q )) R; jj + sup jq(t) ? h(t)j g; t2[T? ;T+ ]


and

9

j j + sup jh (t) ? h(t)j C (e2s T+ + e2u T? + sup j(id ?Q )h(t)j ) t2R

t2[T? ;T+ ] for numbers s and u as in Corollary 2.5.

We point out that super-convergence in the parameter occurs. Indeed, following the proof given in [30], 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 timereversible. Here, we account for this property and adapt the algorithms described above to reversible systems. Consider equation (2.1) u_ = Au + f (u);

(2.9)

u 2 X :

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 , i.e. 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 that a certain transversality condition is satis ed. Hypothesis (H4).

(i) Sh(0) = h(0), i.e. h(0) 2 Fix(S ). (ii) Fix(S ) R(s+ (0; 0)) = X . Here, s+ (t; ) denotes the stable evolution of the variational equation (2.2) about h(t); see Theorem 3.2 below. We then solve the boundary-value problem

u_ = Au + Q f (u); (id ?S )u(0) = 0; R+; (u(T+)) = 0 on the interval [0; T+], where R+; satis es the following hypothesis. Hypothesis (T2). Suppose that X^ is a closed subspace of X and 2 ^ R+; 2 C (X R; X ) is a function such that DR+; and D2 R+; are bounded in a small ball centered at (p ( ); ) in X R uniformly in . Furthermore, Du R+; (p ( ); )jR(P+; ( )) is invertible, and the inverse is bounded uniformly in .

(2.10)

Theorem 2.8. Assume that (A1){(A3), (H1){(H2), (H4), (K), (Q) and (R) are met. Suppose that R+; satis es (T2). The boundary-value problem (2.10) has then a unique solution h for all T+ suciently large and small enough. Furthermore, h (0) 2 Fix(S ) is symmetric, and sup jh (t) ? h(t)j C (jR+; (h(T+ ))j + sup j(id ?Q )h(t)j ): t2[0;T+ ]

t2R

The statements of Theorems 2.1 and 2.7 are also true for (2.10) if adapted appropriately. Corollary 2.9. Assume that (A1){(A3), (H1){(H2), and (R)(i) and (ii) are met. There is then a constant > 0 such that, for any > , (2.9) has a unique

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periodic orbit u near h that has minimal period 2 and satis es u (0) = Su (0). Furthermore, s

sup ju (t) ? h(t)j C e ;

t2[?; ]

where the constant s is as in Corollary 2.5. Proof. We apply Theorem 2.8 with R+ (u) = 21 (id ?S )u and X^ = 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 )jR(P+ ) = idR(P+ ) and (id ?S )P+jR(id ?S) = idR(id ?S) . Hence, Hypothesis (T2) is satis ed. It follows as in [33] that the solution of this boundaryvalue 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, i.e. solutions connecting two different equilibria p as t ! 1. Here, we brie y outline the necessary changes. Suppose that p are hyperbolic equilibria of (2.1) that satisfy Hypothesis (H1). Furthermore, assume that h(t) satis es (H2) but with limt!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 [27, Corollary 1], the adjoint variational equation (2.3)

v_ = ?(A + Du f (h(t); 0) )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 Rm and assume that the m m matrix M with R 1 entries Mij := ?1 h i (t); Dj f (h(t); 0)i dt is invertible.

Note that Hypothesis (H5) is automatically met if m = 0, i.e., if the heteroclinic orbit is transversely constructed. With Hypothesis (H3) replaced by (H5), Theorem 2.1 remains true. Let p; () denote the perturbed equilibria for (2.4). We denote by P? and P?; () the spectral projections of A +Du f (p? ; 0) and A + Q Du f (p?; (); ), respectively, onto the stable eigenspaces corresponding to eigenvalues with negative real part. Similarly, P+ and P+; () are the spectral projections of A + Du f (p+ ; 0) and A + Q Du 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 Du R; (p; ( ); )jR(P; ( )) is invertible, and that the inverse is bounded uniformly in . If Hypothesis (T1)(ii) is replaced by this assumption, the results in the previous sections (except for those pertaining to the existence of periodic solutions) remain true. 3. Exponential Dichotomies | an Excursion. Here, we summarize the results in [27] which are the key to the proofs of the theorems presented in the last section. Assume that the operator A is as in x2, i.e., A : D(A) X ! X is a closed operator such that its domain D(A) is dense in X . Furthermore, A satis es Hypotheses (A1) and (A2). Moreover, let B 2 C 0;# (R; L(X ; X )) be a Holder-continuous


11

family of operators for some # > 0. 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? . Definition 3.1. (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 ) de ned 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) ks (t; )k + ku(; t)k C e?(t? ) for all t in J . (iv) s (t; )s (; s) = s (t; s) for all t s in J , and u (t; )u (; s) = u (t; s) for all t s in J . Note that the operators P (t) = s (t; t) are projections. We assume forward and backward uniqueness of solutions to (3.1) on the interval R. Hypothesis (D). The only bounded solution v(t) of (3.1) on the intervals R+ or R? with v(0) = 0 is the trivial solution v(t) = 0. Similarly, the only bounded solution w(t) of the adjoint equation w_ = ?(A + B (t)) w on R+ or R? with w(0) = 0 is w(t) = 0. Provided B (t) is small for large t, equation (3.1) then has an exponential dichotomy on R+ . Theorem 3.2 ([27]). Suppose that Hypotheses (A1){(A2) and (D) are satis ed. There is then a constant > 0 such that (3.1) has an exponential dichotomy on R+ provided there is a t > 0 such that kB (t)kL(X ;X ) for all t t . Furthermore, the following is true. (i) The projections P (t) = s (t; t) are Holder continuous in t 2 R+ with values in L(X ). (ii) The operators s (t; ) can be extended to operators in L(X ) that satisfy s (t; )s (; s) = s (t; s) for all t s 0. (iii) s (t; ) 2 L(X; X ) for t > and ks (t; )kL(X;X ) C (t ? )? e?(t? ). Analogous properties hold for u (; t) with t 0. The same result is true with R+ replaced by R? . We denote the evolution operators by s+ (t; ) and u+ (; t) de ned for t 0 and by s? (; t) and u? (t; ) de ned for t 0. Finally, we compare the evolution operators for two dierent equations. Lemma 3.3 ([27]). Suppose that B1 (t) satis es the assumptions of Theorem 3.2, including the smallness hypothesis for large values of t, on J = R+ . There exist positive numbers C and 0 such that the following is true. If B2 (t) is such that sup kB1(t) ? B2 (t)kL(X ;X ) < t0

for some < 0 , then the projections Pj (t) with j = 1; 2 that correspond to the equations v_ = (A + Bj (t))v satisfy the estimate

sup kP1 (t) ? P2 (t)kL(X ) C: t0

12


4. The Galerkin Approximation. In this section, Theorem 2.1 is proved. Recall that, throughout this article, C denotes various dierent constants all independent of T? and T+ . We 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 exist a linear, bounded and invertible operator L : Y ! Y^ , an element y0 2 Y , and numbers > 0 and 0 < < 1 such that (i) k id ?L?1DG(y)k for all y 2 B (y0 ), (ii) jL?1G(y0 )j (1 ? ). There exists then a unique point y 2 B (y0 ) with G(y ) = 0, and jy0 ? y j (1 ? )?1 jL?1 G(y0 )j; kDG(y)?1 k (1 + ) kL?1k; uniformly in y 2 B (y0 ).

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. If (Q) is satis ed and K 2 L(X ; X ) is a compact operator, then k(id ?Q )K kL(X ;X ) ! 0 as ! 0. Proof. We argue by contradiction. Suppose that there are elements vn 2 X and n > 0 with jvn j = 1 and n ! 0 as n ! 1 such that j(id ?Qn )Kvn j0 > 0. After choosing a subsequence, we have Kvn ! w in X since K is compact. Hence,

j(id ?Qn )Kvn j0 j(id ?Qn )wj0 + j(id ?Qn )(Kvn ? w)j0 j(id ?Qn )wj0 + C jKvn ? wj0 ?! 0 as n ! 1 due to (Q)(ii). This is a contradiction. Note that Du f (u; 0) is compact for any u provided Hypothesis (K) is satis ed. In this case, (id ?Q )Du f (u; ) converges to zero in the operator norm in L(X ; X ) as and tend to zero. Throughout the remainder of this section, we will make frequent use of this fact. 4.1. Persistence of the Equilibrium. To show that the equilibrium persists, consider (A + Du f (p0 ; 0))?1 (A(p0 + u) + Q f (p0 + u; )) = u + (A + Du f (p0 ; 0))?1 Q (f (p0 + u; ) ? f (p0 ; 0) ? Du f (p0 ; 0)u) +(id ?Q )(f (p0 ; 0) + Du f (p0 ; 0)u) =: G (u; ):

It suces to seek zeros of G (u; ) near (p0 ; 0). The map G is smooth in (u; ) as a map from X R to X and satis es G0 (0; 0) = 0 as well as Du G0 (0; 0) = id. Furthermore, using Ap0 + f (p0; 0) = 0,

jG (0; )j C j(A + Du f (p0 ; 0))?1 (Q (f (p0 ; ) ? f (p0 ; 0)) + (id ?Q )f (p0 ; 0))j C (jj + j(A + Du f (p0 ; 0))?1 A(id ?Q )p0 j ) C (jj + j(id ?Q )p0 j ) and, due to (K) and Lemma 4.2,

kDu G(u; ; ) ? id k C kDu f (p0 + u; ) ? Du f (p0 ; 0) + (id ?Q )Du f (p0 ; 0)k < 21

13


for all (u; ; ) in a ball in X R2 centered at the origin with suciently small radius . We apply Lemma 4.1 for every (; ) in B (0) R2 with L = id. Hence, there exists a unique zero p () 2 B (p0 ) X of G (; ), with p0 (0) = p0 and

jp () ? p0 j C (j(id ?Q)p0 j + jj): Furthermore, p () is smooth in . By construction, p () are equilibria of (2.4). By Hypothesis (K), Theorem 3.2, and Lemma 4.2, the equation

v_ = (A + Q Du f (p (); ))v has an exponential dichotomy on R with projections P+; () and P?; () for all close to zero. This proves the rst and part of the third claim in Theorem 2.1. 4.2. 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), i.e. u_ = Au + Q f (u; ), in the form (4.2)

v_ = (A + Du f (h(t); 0))v + F (t; v; ) = (A + Du f (h(t); 0))v + D f (h(t); 0) + F^ (t; v; )

with F^ (t; v; ) := ?(id ?Q ) Du f (h(t); 0)v + D f (h(t); 0) + f (h(t); 0)

+Q f (h(t) + v; ) ? f (h(t); 0) ? Du f (h(t); 0)v ? D f (h(t); 0) : Due to Hypothesis (K) and Lemma 4.2, we have the estimate (4.3) kD(u;) F^ (t; v; )kL(X ;X ) C (jvj + jj) + g() for some g() with g() ! 0 as ! 0. On account of Theorem 3.2 and Hypotheses (A1){(A3), we know that (2.2)

v_ = (A + Du f (h(t); 0))v has exponential dichotomies on R+ and R? . As in Theorem 3.2, we denote the solution operators of this equation by s+ (t; ) and u+ (; t) for t 0, and by s? (; t) and u? (t; ) for t 0. Solutions of the nonlinear equation (4.2) are bounded on R if, and only if, there exist (b+ ; b?) 2 R(s+ (0; 0)) R(u? (0; 0)) such that

v+ (t) = s+ (t; 0)b+ +

Zt Z0

s+ (t; )F (; v+ ( ); ) d +

Zt

Z1

u+ (t; )F (; v+ ( ); ) d;

t t v? (t) = u? (t; 0)b? + u? (t; )F (; v? ( ); ) d + s? (t; )F (; v? ( ); ) d; 0 ?1 v+ (0) = v? (0); 0 = h'; v+ (0)i:

14


Here, ' 2 (X ) is chosen such that h'; h_ (0)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+ and 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 [27, Lemma 3.1]. Let

2Gv (b(t+);?b?; sv+(t;; v0)?b; )?:=R t s (t; )F (; v ( ); ) d ? R t u (t; )F (; v ( ); ) d 3 66v +(t) ? u+(t; 0)b +? R t0u+(t; )F (; v +( ); ) d ? R t1 +s (t; )F (; v+ ( ); ) d 77 ? ? ? R? 0 ? ?1 ? 7 66 ? R 1 0 u s 4 b+ ? b? ? 0 + (0; )F (; v+R( ); ) d ? ?1 ? (0; )F (; v? ( ); ) d 75 h'; s (0; 0)b ? 1 u (0; )F (; v ( ); ) d i +

+

+

0

+

and consider G : Y ! Y^ for xed as a map de ned on the spaces

Y := R(s+ (0; 0)) R(u? (0; 0)) C 0 (R+ ; X ) C 0(R? ; X ) R; Y^ := C 0(R+ ; X ) C 0 (R? ; X ) X R: Note that G is well de ned and smooth in (b+ ; b?; v+ ; v? ; ). We exploit the splitting (4.2) of F into the linear term D f (h(t); 0) and the quadratic term F^ ; see (4.3). Therefore, consider

G (b+ ; b?; v+ ; v? ; ) = L(b+; b? ; v+ ; v? ; ) 2 R t s (t; )F^ (; v ( ); ) d + R t u (t; )F^ (; v ( ); ) d 66 R t0u+(t; )F^(; v?+( ); ) d + R t1 +s (t; )F^ (; v+? ( ); ) d ? ? 66 R 10 ?u R?1 0 s (0; )F^ (; v ( ); ) d ^ (0 ; ) F ( ; v ( ) ; ) d + 4 0 + ? ?1 ? R+ h'; 01 u+ (0; )F^ (; v+ ( ); ) d i

3 77 77 ; 5

where the linear part L : Y ! Y^ is bounded and given by

2Lv(b+(t;)b??;vs+(; t;v?0); b) =? (R t s (t; )D f (h( ); 0) d + R t u (t; )D f (h( ); 0) d ) 3 66v +(t) ? u+(t; 0)b +? (R t0u+(t; )D f (h( ); 0) d + R t1 +s (t; )D f (h( ); 0) d )77 ? ? ?1 ? 0 ? 7: 66 ? R R 0 1 s u 4 b+ ? b? ? ( 0 + (0; )D f (hR( ); 0) d + ?1 ? (0; )D f (h( ); 0) d ) 75 h'; s (0; 0)b ? 1 u (0; )D f (h( ); 0) d i +

+

0

+

Note that the last two components of L do not depend on (v+ ; v? ). The linear operator 0 : R(s+ (0; 0)) R(u? (0; 0)) ?! X ;

0 (b+ ; b? ) = b+ ? b?

is a Fredholm operator with index zero. Its null space and range are given by N(0 ) = spanf(h_ (0); h_ (0))g R(s+ (0; 0)) R(u? (0; 0)); R(0 ) = fv 2 X ; h (0); vi = 0g: On the other hand,

h'; s+ (0; 0)h_ (0)i = h'; h_ (0)i = 1;


and

Z1 ?1

=

D

h ( ); D f (h( ); 0)i d (0);

Z1 0

u+ (0; )D f (h( ); 0) d +

Z0 ?1

15

E

s? (0; )D f (h( ); 0) 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 to the map G for every small xed with y0 = (0; 0; 0; 0). Hence, we obtain the existence and uniqueness statements in (ii) and (iii) of Theorem 2.1. 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

jG (0; 0; 0; 0)jY^ C sup j(id ?Q )h(t)j

(4.4)

t2R

where

= 2G (0R; t0; 0s ;(0t;; 0) )(id R ?Q )f (h( ); 0) d + 1t u+ (t; )(id ?Q )f (h( ); 0) d + 0 66 R t u (t; )(id ?Q )f (h( ); 0) d + R t s (t; )(id ?Q )f (h ); 0) d ?1 ? 66 R 10 u? R 0 s 4 0 + (0; )(id ?Q)Rf (h( ); 0) d + ?1 ?(0; )(id ?Q)f (h( ); 0) d h'; 1 u (0; )(id ?Q )f (h( ); 0) d i 0

+

3 77 77 : 5

In order to prove (4.4), it suces to show that

Z t s j(id ?Q )h( )j 0 + (t; )(id ?Q)f (h( ); 0) d C sup 2R

(4.5)

for some constant C independently of t 0, and similar estimates for the other integrals. We use the fact that h(t) satis es h_ = Ah + f (h; 0) for t 2 R, i.e.

h 2 C 1 (R; X ) \ C 0 (R; D(A)):

(4.6)

Therefore, using also (Q)(i), we have

Zt

Zt

s+ (t; )(id ?Q ) dd h( ) ? Ah( ) d 0Z 0 t d s = ? d + (t; ) (id ?Q )h( ) ? s+ (t; )A(id ?Q )h( ) d 0 +s+ (t; t)(id ?Q )h(t) = =

s+ (t; )(id ?Q )f (h( ); 0) d =

Z t

s+ (t; )(A + Du f (h( ); 0))(id ?Q )h( ) ? s+ (t; )A(id ?Q )h( ) d 0 +s+ (t; t)(id ?Q )h(t)

Zt 0

s+ (t; )Du f (h( ); 0)(id ?Q )h( ) d + s+ (t; t)(id ?Q)h(t):

Note that integrating by parts and taking the derivative dd s+ (t; ) is allowed on account of (4.6). Using Theorem 3.2(iii), it is now straightforward to obtain the

16


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 that the homoclinic orbits h (t) of (2.4) are nondegenerate. Lemma 4.3. The only bounded solution, up to constant multiples, of the variational equation v_ = (A + Q Du f (h (t); )v about h (t) is given by h_ (t). In other words, the solutions h (t) are nondegenerate. Proof. On account of Theorem 3.2, the variational equation v_ = Av + Q Du f (h (t); )v has exponential dichotomies on R+ and R? with solution operators s+; (t; ) and u+; (; t) for t 0, and s?; (; t) and u?; (t; ) for t 0. Due to Hypothesis (K), and Lemmata 3.3 and 4.2, s+; (0; 0) and u?; (0; 0) are close to s+ (0; 0) and u? (0; 0), respectively, in the L(X )-norm. Therefore, h_ (t) is the only bounded solution, up to constant multiples, of (2.4). It is a consequence of the proof of Lemma 4.3 that h_ 2 C 0 (R; X ). Indeed, _h (0) 2 R(s+; (0; 0)), and therefore h_ (0) 2 X . Furthermore, h_ (t) = s+; (t; 0)h_ (0) for t > 0 is continuous in t as a function into X by Theorem 3.2. Since the choice of t = 0 is arbitrary, we see that in fact h_ 2 C 0 (R; X ). 5. The Truncated Boundary-Value Problem. In this section, we prove Theorem 2.3. Again, C denotes various dierent constants that are independent of T? and T+ . 5.1. The Nonlinear Equation. We exploit the transformation u(t) = h(t) + v(t) and = + . The function v(t) then satis es the equation (5.1) v_ = (A + Du f (h(t); 0))v + F (t; v; ) = (A + Du f (h(t); 0))v + D f (h (t); ) + F^ (t; v; ); where F^ (t; v; ) = Q f (h (t) + v; + ) ? f (h (t); ) ? Du f (h(t); 0)v ?(id ?Q )Du f (h(t); 0)v ? D f (h (t); ): The derivative D(v; ) F^ (t; v; ) is given by D(v; ) F^ (t; v; ) =

h

Q (Du f (h (t) + v; + ) ? Du f (h(t); 0)) ? (id ?Q )Du f (h(t); 0); i Q (D f (h (t) + v; + ) ? D f (h (t); )) ? (id ?Q )D f (h (t); ) :

Due to Theorem 2.1, Hypothesis (K), and Lemma 4.2, we obtain the estimate (5.2) kD(v; ) F^ (t; v; )kL(X R;X ) C (jvj + j j) + g() uniformly in for (v; ) in a ball centered at zero of suciently small radius in X R. Here, g() ! 0 as ! 0. Let (5.3) a = (a+ ; a? ) 2 Xa := R(P+ ) R(P? ); b = (b+ ; b? ) 2 Xb := R(s+ (0; 0)) R(u? (0; 0)):


17

We de ne the maps

I+;T; : Xa Xb C 0 ([0; T+ ]; X ) R ?! C 0 ([0; T+]; X ); I?;T; : Xa Xb C 0 ([T? ; 0]; X ) R ?! C 0 ([T?; 0]; X ) by (5.4)

I+;T; (a; b; v+ ; )(t) = u+ (t; T+ )a+ + s+ (t; 0)b+ +

Zt

T+

u+ (t; )F (; v+ ( ); ) d +

Zt 0

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) satis es the integral equation (5.5)

0 = v+ (t) ? I+;T; (a; b; v+ ; )(t); 0 = v? (t) ? I?;T; (a; b; v? ; )(t)

together with the equation v+ (0) = v? (0) for some (a; b). Here, t 2 [0; T+] in the rst, and t 2 [T? ; 0] in the second equation in (5.5). In addition, we have to solve the phase and boundary conditions (5.6)

R (h (T+ ) + v+ (T+ ); h (T?) + v? (T? ); + ) = 0; JT; (h + V (v+ ; v? ); + ) = 0;

where the linear, bounded operator

V : C 0 ([0; T+]; X ) C 0 ([T? ; 0]; X ) ?! C 0 ([T? ; T+]; X ) is de ned by (5.7)

v (t) + v (0) ? v (0) ? + V (v+ ; v? )(t) = + v? (t)

t > 0; t 0:

For v 2 X , we expand the boundary conditions according to (5.8) R (h (T+ ) + v+ ; h (T? ) + v? ; + ) = R (h (T+ ); h (T? ); ) +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 + j j) for (v+ ; v? ; ) in a ball with small radius centered at zero in X X R, independently of . Similarly, for v 2 C 0 (T; X ), we have

(5.9)

(5.10)

JT; (h + v; + ) = JT; (h ; ) + Dv JT; (h ; )v +D JT; (h ; ) + J^T; (h ; v; )

with (5.11)

kD(v; ) J^T; (h ; v; )k C (jvj + j j)

18


for (v; ) in a ball with small radius centered at zero in C 0 (T; X ) R, independently of . We consider the nonlinear equation (5.12) GT; : Y ?! Y^ ; GT; (a; b; v+ ; v? ; ) = 0 with

Y = Xa Xb C 0 ([0; T+]; X ) C 0([T? ; 0]; X ) R Y^ = C 0 ([0; T+]; X ) C 0([T? ; 0]; X ) X X R de ned by the right-hand side of (5.5), the continuity equation 0 = I+;T; (a; b; v+ ; )(0) ? I?;T; (a; b; v? ; )(0); the rst two equations in (5.6), and the equation 0 = JT; (h + V (I+;T; (a; b; v+ ; ); I?;T; (a; b; v? ; )); + ): It is a consequence of the above discussion that G is well de ned and smooth. Furthermore, due to the estimates (5.2), (5.9) and (5.11), we can solve (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? ; ) = 0 is invertible uniformly in T and . The arguments are analogous to those presented in the last section, and we omit them. Note that the error estimate in Theorem 2.3 follows from Lemma 4.1 since

GT; (0) = 0; 0; 0; R(h (T+ ); h (T? ); ); JT; (h ; ) : Upon replacing h () by h ( + T; ) for an appropriate small number T; , we can achieve that JT; (h ; ) = 0. 5.2. The Linearized Boundary-Value Problem. It remains to show that the operator LT; = DGT; (0) is invertible as a map from Y to Y^ . Let (5.13) I^+;T; (a; b; )(t) = u+ (t; T+)a+ + s+ (t; 0)b+ +

Z t

T+

u+ (t; )D f (h ( ); ) d +

Zt 0

s+ (t; )D f (h ( ); ) d ;

(5.14) I^?;T; (a; b; )(t) = s? (t; T? )a? + u? (t; 0)b? +

Z t

T?

s? (t; )D f (h ( ); ) d +

Zt 0

u? (t; )D f (h ( ); ) d

for t 2 [0; T+] and t 2 [T?; 0], respectively. The linear operators I^ are bounded from Xa Xb R into C 0 ([0; T+]; X ) and C 0 ([T? ; 0]; X ), respectively. We then have (5.15) LT; (a; b; v+ ; v? ; ) =

2 v+ ? I^+;T; (a; b; ) 66 v? ? I^?;T; (a; b; ) 66 ^ ? I^?;T; (a; b; )(0) 64 DR (h (T+); hI(T+?;T;); (a; )b;I^+)(0) ;T; (a; b; )(T+ ); I^?;T; (a; b; )(T? );

Dv JT; (h ; )V (I^+;T; (a; b; ); I^?;T; (a; b; )) + D JT; (h + v; )

3 77 77 : 75


19

We have to show that the equation

LT; (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? ; )jY C j(g+ ; g? ; c; r; j )jY^ for some positive constant C that is independent of and T . Inspecting the de nition (5.15) of LT; , it is clear that we can solve the rst two components in (5.16) for (v+ ; v? ) such that (v+ ; v? ) = W1 (a; b; ; g+; g? ) with kW1 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+ ; 0)b+ + w? = s? (T?; T? )a? + u? (T? ; 0)b? +

Z0 ZT+0 T?

s+ (T+ ; )D f (h ( ); ) d; u? (T? ; )D f (h ( ); ) d:

The key is that D(u+ ;u? ) R (h (T+ ); h (T? ); )(u+ (T+ ; T+)jR(P+ ) ; s? (T?; T? )jR(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 suciently large and small enough due to Hypothesis (K) and Lemmata 3.3 and 4.2. Therefore, we can solve (5.17) for a = (a+ ; a? ) and obtain a = W2 (b; ; r) with kW2 k C independently of T? , T+ and . Actually, we obtain the better estimate

jW2 (b; ; r)j C (e?T+ jb+ j + eT? jb? j + j j + jrj ) using the estimate

js+ (T+ ; 0)b+ j C e?T+ jb+ j and an analogous estimate for u? (T? ; 0); see Theorem 3.2. Next, we apply these estimates to the operator V (I^+;T; (a; b; ); I^?;T; (a; b; )) appearing in the phase condition; see (5.7) for its de nition. Using (5.13), (5.14) and the estimates for a, we obtain the expansion

V (I^+;T; ; I^?;T; )(a; b; )(t) =

s (t; 0)b

+ + b? ? b+ + W3 (b; ; r)(t) + u? (t; 0)b? + W3 (b; ; r)(t)

with

jW3 (b; ; r)(t)j C (e?T+ jb+j + eT? jb?j + j j + jrj ):

t > 0; t0

20


According to the results in x4.2, we may write (b+ ; b? ) = (^b+ ; ^b?) + (h_ (0); h_ (0));

(^b+ ; ^b?) 2 X^b

with X^b spanf(h_ (0); h_ (0))g = Xb. We obtain V (I^+;T; ; I^?;T; )(a; b; )(t) = h_ (t) + W4 (b; ; r)(t) with

jW4 (b; ; r)(t)j C (e?jT jj j + j^b+j + j^b? j + j j + jrj ) where jT j := minfjT?j; T+g. The phase condition and continuity equation are then given by

j = Dv JT; (h ; )h_ + W5 (b; ; r); (5.18) c = u+ (0; T+)W2;+ (b+ ; ; r+ ) ? s? (0; T?)W2;? (b? ; ; r? ) + ^b+ ? ^b?

? with

Z T+ 0

u+ (0; )D f (h ( ); ) d +

Z0

T?

s? (0; )D f (h ( ); ) d

jW5 (b; ; r)j C (e?T+ jb+ j + eT? jb?j + j j + jrj ):

Note that we have the estimates

ju+ (0; T+)W2;+ (b+ ; ; r+ ) ? s? (0; T? )W2;? (b? ; ; r? )j C (e?T+ + eT? )(jb+ j + jb?j + j j + jrj ) due to Theorem 3.2. Moreover,

Z T+ T?

h (t); D f (h (t); )i dt

is bounded away from zero due to Hypothesis (H3), Theorem 2.1 and the fact that (t) converges to zero exponentially. Therefore, using Theorem 2.1 and Hypothesis (T1)(i), and arguing as in x4.2, we can solve (5.18) for (^b; ; ). 5.3. Proof of Theorem 2.8. The proof is a consequence of the proof of Theorem 2.3. 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(0) = 0 due to the transversality condition (R)(ii). We omit the details. 6. The Finite-Dimensional Boundary-Value Problem. In this section, we prove Theorem 2.7. We embed the boundary-value problem on R(Q ) into a larger one de ned on X and then apply Theorem 2.3. Any element u in X can be written according to

u = q + w;

(q; w) 2 R(Q ) N(Q ):

Using this decomposition, we have (A + Q D f (p (); ))j Q D f ( p ( ) ; ) j u u R( Q ) N( Q ) : A + Q Du f (p (); ) = 0 AjN(Q )


21

The spectral projections of A, A + Q Du f (p (); ) and (A + Q Du f (p (); ))jR(Q ) are denoted by P^ , P; () and Q; (), respectively; see Hypotheses (A1) and (H1). On account of Hypothesis (Q), we then have

Q () ; P; () = 0

D; () (id ?Q )P^

for some bounded operators D; (). The equation u_ = Au + Q f (u; ) is equivalent to (6.1)

q_ = Aq + Q f (q + w; );

w_ = Aw:

We include the phase and boundary conditions

(6.2)

Z T+

hh_ (t); q(t) + w(t) ? h (t)iX dt; T? R~+ ((q; w)(T+ ); ) = P+; ( )P+; ()(q(T+ ) + w(T+ ) ? p ()); R~? ((q; w)(T? ); ) = P?; ( )P?; ()(q(T? ) + w(T? ) ? p ()): J~((q; w); ) =

We rst prove that (6.1{6.2) has a unique solution. Using Remark 2.2, it is straightforward to show that (6.2) satis es Hypothesis (T1). For instance, for = ,

Du R~+ (p ( ); )

R(P; ( ))

= 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 2.3 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 = 0. Indeed, w has to satisfy w_ = Aw; P^+ w(T+ ) = 0; P^? w(T? ) = 0: Since AjN(Q ) is hyperbolic, w = 0 is the only solution. With w = 0, it is easy to see that (6.1{6.2) and (2.6{2.8) coincide. Hence, (q; w) = (q; 0) satis es (6.1{6.2) if, and only if, q is a solution of (2.6{2.8). Finally, we have

jR+; (h(T+ ); 0)j C jh(T+ )j2 C e2s T+ ; and the analogous estimate for R?; (h(T? ); 0). This completes the proof of Theorem 2.7. 7. Semilinear Elliptic Equations. Here, we show that elliptic equations on in nite cylindrical domains are included in the abstract set-up of the earlier sections; for more details, we refer to [27]. Furthermore, we comment on the satisfaction of the hypotheses of Theorems 2.1 and 2.3, and discuss particular discretizations. Let Y be a Hilbert space and L : D(L) Y ! Y be a densely de ned, positive de nite, self-adjoint operator with compact resolvent. In most applications, we have Y = L2 ( ) for some bounded domain and L = ? on together with Dirichlet

22


boundary conditions, say, so that D(L) = H 2 ( ) \ H01 ( ). 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 [0; 1) and > 0. Consider the abstract

elliptic equation

uxx ? Lu = g(u; ux);

(7.1)

x2R

for u 2 Y . We reformulate (7.1) as the rst-order equation

d u = A u + G(u; v) dx v v

(7.2)

with (u; v) = (u; ux) and G(u; v) = (0; g(u; v)). Here,

A=

0 id L 0

: Y 1 Y 12 ! Y 21 Y:

In particular, Hypothesis (A1) is met; in fact, the projections P^ are given by

P^ = 21

id 1 L? 21 L 2 id

: Y 21 Y ?! Y 12 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. Thus, Hypotheses (A1), (A2) and (K) are met. We refer to [9, Satz 5] for conditions that guarantee 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; ux; ry 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 N g 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 is in general less evident as the projectors P+; and P?; might be hard to compute. We emphasize that, in general, simple Dirichlet boundary conditions u(T ) = p or Neumann boundary conditions v(T ) = 0 will not work. Even for systems of equations on the line with no cross-section, i.e. for Y = R2k , the dimensions of stable and unstable subspaces at the equilibrium may not coincide, i.e. dim R(P+ ) 6= k, and Dirichlet as well as Neumann boundary conditions yield ill-posed problems. The only generic choice 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 that we discuss next in more detail. If g = g(u), the system is reversible. Reversibility acts through S (u; v) = (u; ?v). The condition (id ?S )u(0) in (2.10) reduces to v = 0, in other words, to Neumann boundary conditions at x = 0. The hyperbolicity assumption (H1) is then equivalent


23

to linear stability of the equilibrium u(x; y) = p(y) for the parabolic equation ut = uxx ? Lu ? g(u) on the cylinder. Due to the second-order structure, p eigenfunctions of the linearization of (7.1) at the equilibrium are of the form (uk ; k uk ) where k and uk are eigenvalues and eigenfunctions, respectively, of L + Dg(p). By hyperbolicity, k > 0. We claim that we can choose Dirichlet or Neumann conditions at x = T+ as p well. Indeed, the stable subspace R(P? ) is spanned by (uk ; k uk ) and the spaces f(u; v); u = 0g or f(u; v); v = 0g are closed complements of this subspace. We

summarize this discussion in the following proposition. Proposition 7.1. Assume that (H1), (H2) and (H4) are met. Furthermore, suppose that g = g(u). Dirichlet and Neumann boundary condition then satisfy (T2). These arguments can be slightly generalized to elliptic equations with variational structure

uxx = Lu + cux + rF (u); where heteroclinic orbits that connect stable equilibria are of interest. Again, stability is with respect to the linearization of the associated parabolic problem in the in nite cylinder. Though this system is not reversible, a calculation similar to the one given above shows that Dirichlet or Neumann boundary conditions at x = T? and x = T+ satisfy Hypothesis (T1)(ii) on the boundary conditions. We remark that Corollaries 2.6 and 2.9 establish the existence of solutions of (7.1) that are periodic in x with arbitrarily large period and that have the same pro le in the cross-section as the solitary wave.

8. Numerical Simulations. In this section, we compare the theoretical predictions with numerical computations. Consider the elliptic equation (8.1) uxx + uyy + cux = u(1 + 2p ? u) + pyy ? p(1 + p);

(x; y) 2 R (?1; 1)

for u 2 R with Neumann boundary conditions

uy (x; 1) = 0;

(8.2)

x 2 R:

For the function p(y), we take the polynomial p(y) = (1 + y)2 (1 ? y)2 which clearly satis es py (1) = 0. Note that p(y) satis es (8.1{8.2) for any c. Furthermore,

h(x; y) = p(y) + 23 sech2 x2

(8.3)

is an explicit solitary wave of (8.1) for c = 0. We write (8.1) as the rst-order system (8.4)

v d u = dx v ?uyy ? cv + u(1 + 2p ? u) + pyy ? p(1 + p)

in x, where (u; v) 2 H 1 (?1; 1) L2 (?1; 1). It turns out that Hypotheses (A1){(A3), (H1){(H3) and (K) are satis ed with respect to the parameter c; see x7 and [27, x6.3]. Alternatively, we may x c = 0. Equation (8.4) is then reversible with S (u; v) = (u; ?v), and Hypotheses (R)(i-ii) and (H4) are met.

24


8.1. Projection Boundary Conditions. The even eigenfunctions and corresponding eigenvalues of the linearization of (8.4) at (p; 0) are given by (1 + 2k2)? 12 p 22 cos ky; = (8.5) qk (y) = 1+ k ; k 1 respectively, for k 2 N 0 . We consider the Galerkin projection Qn =

(8.6)

n X

k=?n

hqk ; iH 1 L2 qk ;

which clearly satis es Hypothesis (Q). The Fourier series of the polynomial p(y) is given by 1 (?1)k+1 16 + 48 X p(y) = 15 4 k=1 k4 cos ky;

and j(id ?Qn)(p; 0)jL2 L2 n?9=2 (an + b) (8.7) for some positive numbers a and b. We then solve the system d u =Q v n ?u ? cv + u(1 + 2p ? u) + p ? p(1 + p) ; dx v yy yy

ZTD

E

Qn(hx ; hxx )(x); (u; v)(x) ? Qn (h; hx )(x) L2 L2 dx; ?T 0 = Q+;n(c) (u; v)(T ) ? (pn (c); 0) ;

(8.8)

0=

0 = Q?;n(c) (u; v)(?T ) ? (pn (c); 0)

n9=2 (T; n)

450 350 250 150 50

2

6

10

n

14

18

22

Fig. 1. This plot contains the scaled error n9=2 (T; n) of the solution to (8.8) versus the number n of Galerkin modes for a xed length T = 15:0 of the truncation interval. In this scaling, the error is a linear function of n; see (8.9).


25

0

log (T; n)

-2 -4

n=8 n = 12 n = 16 n = 20 n = 25

-6 -8 -10

1

2

3

T

4

5

6

n=1

Fig. 2. The scaled error log (T; n) of the solution to (8.8) versus the length T of the interval (?T; T ) is shown. For small T , the scaled error is linear in T with slope ?1:8; this is in agreement with the prediction of ?2 in Theorem 2.7. For larger values of T , the error due to the Galerkin truncation becomes dominant. As expected, this remaining error is smaller for a larger number of modes. Also, since the error curves for dierent values of n are not shifted against each other, the picture con rms that the constant C appearing in (8.9) is independent of n. The case n = 1 corresponds to setting p(y) = 0 which demonstrates the error induced purely by the truncation of the interval.

on (?T; T ) with (u; v) 2 R(Qn ); see x2.4. Since Hypothesis (T1) is met, Theorem 2.7 is applicable. Therefore, the dierence (T; n) of the true solution h(x; y) given in (8.3) and the solution h n (x; y) of (8.8) can be estimated by (T; n) = sup fjh n (x; ) ? h(x; )jL2 L2 g (8.9) x2(?T;T ) C (e?2T + j(id ?Qn )(p; 0)jL2 L2 ) C (e?2T + n?9=2 (an + b));

using the expression for the eigenvalues given in (8.5). The boundary-value problem (8.8) is now solved using Auto97, see [10], for various choices of T and n. The results of the numerical simulations are plotted in Figures 1 and 2. They con rm the theoretical error estimate (8.9). Note that the error is not determined by the residual (id ?Qn )(pyy ? p(1 + p); 0) on the right-hand side of (8.8) but by the approximation of the true solution using Galerkin modes. 8.2. Neumann Boundary Conditions. Next, we consider the approximation uxx + uyy = u(1 + 2p ? u) + pyy ? p(1 + p); (x; y) 2 (0; T ) (?1; 1); (8.10) uy (x; 1) = 0; x 2 (0; T ); ux(0; y) = ux(T; y) = 0; y 2 (?1; 1): Hypothesis (T2) is met, and Theorem 2.8 applies with Q id. Hence, the dierence (T ) of the solution h(x; y) given in (8.3) and the solution h (x; y) of (8.10) can be estimated by (8.11) (T ) = supfjh (x; y) ? h(x; y)j; (x; y) 2 (?T; T ) (?1; 1)g C e?T

26

G. LORD, D. PETERHOF, B. SANDSTEDE, AND A. SCHEEL -2

log (T; n)

-3 -4 -5

N = (200; 50)

-6

N = (400; 150)

-7 -8

N = (600; 150)

4

5

6

7

8

T

9

10

Fig. 3. The scaled error log (T; N ) of the solution to (8.10) versus the length T of the interval (?T; T ) is shown. Here, N = (Nx ; Ny ) is the number of horizontal and vertical grid points. For small T , the scaled error is linear in T with slopes of ?0:86, ?0:99 and ?1:01. The slope predicted in Theorem 2.8 is ?1. If T is large enough, the error due to the approximation by nite dierences becomes dominant; again, this remaining error is smaller for a larger number of grid points.

-2

log (T; n)

-3 -4 -5 -6

N = (300; 75)

-7 -8

N = (400; 100)

2

3

4

T

5

6

Fig. 4. The error for the solution of (8.12) is plotted. For smaller T , the scaled error is linear in T with slope ?1:86; Theorem 2.8 predicts a slope of ?2.

using the expression for the eigenvalues given in (8.5). We used second-order centered nite dierences on a staggered grid with Nx horizontal and Ny vertical mesh points in order to solve (8.10). For the resulting equation on the grid, we employed a conjugated-gradient solver (without preconditioning) together with Newton's method. The dierence of the associated solution h N and the true solution h is denoted by (T; N ) where N = (Nx ; Ny ). The results of the numerical simulations are shown in Figure 3. Again, the theoretical predictions of Theorem 2.8 are in good agreement with the computations.


27

For comparison, we also computed solutions of

uxx + uyy = u(1 + 2p ? u) + pyy ? p(1 + p); (8.12) uy (x; 1) = 0; ux(0; y) = 0; ux(T; y) + u(T; y) ? p(y) = 0;

(x; y) 2 (0; T ) (?1; 1); x 2 (0; T ); y 2 (?1; 1); y 2 (?1; 1):

These are the projection boundary conditions. The error is therefore expected to behave like e?2T by Theorem 2.8; see Figure 4. 9. An Application to the von Karman{Donnell Equations. As mentioned in the introduction, we consider the post buckling of an in nitely long cylindrical shell under axial compression as modeled by the von Karman{Donnell equations. In [23, 24, 25] solitary-waves were computed, and it was shown that these solutions provide a good approximation to the localized buckling pattern observed in experiments. Here, we indicate how the proofs in x4 and x5 may be adapted to this case and show numerically that, for a xed spatial truncation, the error in the truncation on the length of the cylinder scales in accordance with Theorem 2.8. 9.1. The von Karman{Donnell Equations. The classical formulation for a thin cylindrical shell of radius r and thickness t is given by the von Karman{Donnell equations

2 2 w + wxx ? xx = wxx yy + wyy xx ? 2wxy xy ; 2 + wxx = (wxy )2 ? wxx wyy where 2 is the two-dimensional bi-harmonic operator, x 2 R is the axial and y 2 [0; 2r) is the circumferential coordinate, w is the outward radial displacement measured from an unbuckled state, and is a stress function [21]. Parameters appearing in (9.1) are the curvature = 1=r, the geometric constant 2 = t2 =12(1 ? 2), where is Poisson's ratio, and the loading parameter . Localized buckle patterns are observed, and these are well approximated by a solitary wave in x, see [23, 25]

(9.1)

and Figure 5. We discretize the von Karman{Donnell equations (9.1) in such a way as to exploit the natural symmetries in the problem. Experimentally [21, 35], a well de ned number s of periodic waves is observed circumferentially in the buckled deformation, corresponding to an invariance under rotation of 2=s. Hence, we write

w(x; y) =

1 X

m=0

am (x) cos(msy);

(x; y) =

1 X

m=0

bm(x) cos(msy);

s 2 N:

Substituting into the von Karman{Donnell equations and taking the L2 inner product with cos(msy), we obtain a system of nonlinear ODEs for the Fourier modes am and bm for m = 0; : : : ; 1. The Galerkin approximation is formed by taking m = 0; : : : ; M ? 1 for some nite M giving a system of 8M rst-order ordinary dierential equations. We may formally write the resulting set of ODEs as

aivm = F1;m (am ; aim ; aiim ; aiiim; bm ; bim ; biim; biiim ) bivm = F2;m (am ; aim ; aiim ; aiiim; bm ; bim ; biim; biiim ); where superscripts denote dierentiation with respect to x.

28


Note that s = 1 corresponds to the standard Galerkin approximation. Convergence as M is increased was examined numerically in [24], and it was found that M = 6 gives a reasonable compromise between accuracy and computation eciency. Experimentally, the observed buckle patterns tend to be cross-symmetric about a section x = T=2, i.e. (9.2) w(x; y) = w(T ? x; y + r=s); (x; y) = (T ? x; y + r=s): In terms of Fourier modes, the constraints (9.2) are equivalent to a2k (T ? x) = a2k (x); a2k+1 (T ? x) = ?a2k+1 (x); b2k (T ? x) = b2k (x); b2k+1 (T ? x) = ?b2k+1 (x): This form of solution is depicted in Figure 5 with s = 11 circumferential waves. Symmetric solutions also exist but are not commonly observed. Note also that the equations for the zero mode (m = 0) may be solved for aii0 ; aiii0; bii0; biii0 independently of the initial conditions for a0 ; ai0 ; b0 ; bi0. This corresponds to a trivial translational symmetry in the problem (sometimes called a rigid-body mode). This symmetry introduces four eigenvalues at zero in the linearization about the trivial solution. In computations, projection boundary conditions are imposed at x = 0, while at x = T=2 we impose boundary conditions associated with the cross-symmetric

Fig. 5. De ection w(x; y) (left) and stress function (x; y) (right) reconstructed from the numerical solution with s = 11 circumferential waves and M = 6. See text for values of the other parameters.


29

section de ned by (9.2). For further details on the numerical solution of the resulting boundary-value problem see [24]. Results produced here are for r = 100, s = 11, t = 0:247, = 0:3 and = 4:5 10?4 . 9.2. Convergence Results for the von Karman{Donnell Equations. Before we compare the numerical error with the theoretical prediction, we show that our results are actually applicable to the von Karman{Donnell equations. The symmetries for the Galerkin approximation are induced by the following symmetries of the von Karman{Donnell equations. Equation (9.1) admits the Z2equivariance x 7! ?x, the S 1 -symmetry R : y 7! y + for 2 [0; 2r) and the rigid-body R4 -symmetry (w; ) 7?! (w + d0 + d1 x; + d3 + d4 x) for d 2 R4 . Cross-symmetric solutions having s periodic patterns are contained in the xed-point space of the Zs-action y 7! y + 2r s . Using the dynamical-systems coordinates (9.3) (W; )(x) = (w0 ; w1 ; w2 ; w3 ; 0 ; 1 ; 2 ; 3 )(x); the von Karman{Donnell equations can be written as an abstract rst-order equation @x wj = wj+1 ; j = 0; 1; 2; @x j = j+1 ; j = 0; 1; 2; (9.4) @xw3 = ?@yyyy w0 ? 2@yy w2 + 12 (?w2 + 2 + w2 @yy 0 + 2 @yy w0 ? 2(@y w1 )(@y 1 )); @x3 = ?@yyyy 0 ? 2@yy 2 ? w2 + (@y w1 )2 ? w2 @yy w0 : The functions (W; ) are in the function space 3 H 2 H 1 L2 )([0; 2r))]2 Y = [(Hper per per

3?j . Cross-symmetric solutions are then contained in the such that wj ; j 2 Hper subspace of Y consisting of all functions which are invariant under the Zs-action y 7! y + 2r s . We claim that, with this choice of Y , the von Karman{Donnell equations satisfy the conditions stated in x2. Using explicit calculations on the Fourier coecients, it is straightforward to show that the principal part @x wj = wj+1 ; @x j = j+1 ; j = 0; 1; 2; @x w3 = ?@yyyy w0 ; @x 3 = ?@yyyy 0 satis es the resolvent estimate and the spectral decomposition stated in Hypothesis (A1). Moreover, the resolvent of the right-hand side is compact since its range is 4 H 3 H 2 H 1 )([0; 2r))]2 . Since the other terms appearing in Y 1 = [(Hper per per per (9.4) are of lower order, the linearization of the full equation at an equilibrium state satis es Hypotheses (A1) and (A2) provided the equilibrium is hyperbolic up to the four eigenvalues at zero induced by the symmetry. The uniqueness assumption (A3) is a consequence of analyticity. We refer to [12] for more details on higher-order elliptic equations. We remark that it is possible to check hyperbolicity and other assumptions on the linearization about the origin with explicit calculations using the Fourier expansion.

30


Next, we introduce the reversibility operator S (W; ) 7?! S (W; ) := (Rr=s w1 ; ?Rr=sw2 ; Rr=s w3 ; ?Rr=s w4 ; Rr=s 1 ; ?Rr=s 2 ; Rr=s 3 ; ?Rr=s 4 ) with Rr=s u(y) = u(y + r=s); and the R4 -symmetry (W; ) 7?! d(W; ) := (w1 + d1 ; w2 + d2 ; w3 ; w4 ; 1 + d3 ; 2 + d4 ; 3 ; 4 ); where d = (d1 ; :::; d4 ) 2 R4 . Suppose that (W ; )(x) is a cross-symmetric localized solution of the von Karman{Donnell equations. Using the variables (9.3), we see that S (W ; )(0) = (W ; )(0). This solution then generates a group orbit of solutions d(W ; )(x) that converge to the equilibria (d1 ; :::; d4 ) as x ! 1. The proofs in x4 and x5 can be adapted to this situation by using the parametrization u(t) = d h(t) + v(t) instead of (4.1). Here, d 2 R4 describes the directions along the group orbit. Using this parametrization, it can then be shown that the proofs carry over to the von Karman{Donnell equations. The variable d accounts for the neutral directions associated with the symmetries while we have exponential dichotomies for the variable v at our disposal. We omit the actual calculations. The crucial point is the following. The set f(0; d2 ; 0; d4 )(W ; )(0)g is transverse to Fix(S ), while (d1 ; 0; d3 ; 0)(W ; )(0) 2 Fix(S ) for any d1 ; d3 . Therefore, by imposing the boundary condition (W ; )(0) 2 Fix(S ), we obtain d2 = d4 = 0. We then add the condition (W ; )(T ) ? f(d1 ; 0; d3 ; 0)g at x = T . These boundary conditions, together with projection boundary conditions at x = T , select a unique solution in the group orbit R4 (W ; )(x) of solitary waves. 9.3. Comparison of Numerical Computations and Theory. Theorem 2.8 predicts that the error of the exact solution (W ; ) and the numerical solution (WT ; T ) computed on the interval [0; T ] is given by s sup k(W ; ) ? (WT ; T )kL2 (0;2) C e T : x2[0;T ]

Here, and in the following, we neglect the error due to the Galerkin approximation. Numerically, we obtain s = ?0:0223 for the real part of the eigenvalue closest to the imaginary axis. In order to compare the theoretical prediction, we calculate (WT ; T ) for T = 1000 large and compare it to solutions (WT ; T ) computed on much smaller intervals. The results are shown in Figure 6. The slope of the error curve is about ?0:018 which compares well with the predicted slope of ?0:0223.

p

log( jW ? WT j2 + j ? T j2 )

NUMERICAL COMPUTATION OF SOLITARY WAVES IN INFINITE CYLINDERS -3

-3.5

-4

31

-4.5

-5 300

Fig. 6. The error log(

320

340

T

360

380

400

pjW ? WT j2 + j ? T j2) versus the length T of the shell is plotted.

Here, the \exact" solution (W ; ) has been computed for T = 1000, while the length used for the calculation of the solution (WT ; T ) varies. The slope of the error curve is approximately equal to ?0:018 which compares well with the prediction of ?0:0223.

9.4. Consequences of the Results on Truncation. The arguments given in x9.2 show that the results in x2, and in particular Theorem 2.8, are applicable to the

von Karman{Donnell equations provided a solitary wave exists on the in nite cylinder. Therefore, we can conclude that solutions to the truncated boundary-value problem exist as long as the boundary conditions are admissible. Moreover, the solutions do not depend much on the boundary conditions. This has interesting consequences for the buckling of shells. First of all, the cylinders used in experiments are, of course, of nite length. Our results then show that the buckle patterns do not depend very much on the boundary conditions imposed at the ends of the shell. Whether the shell is simply supported or clamped in a dierent way has only little eect on the buckling pattern, indeed the eect is exponentially small (this is sometimes referred to as the Saint-Venant Principle). However, the stability and positioning on the cylinder of the resulting pattern may depend on the boundary conditions; it is possible that patterns cannot be observed experimentally under certain boundary conditions. For instance, the stability of localized buckling states of a strut resting on an elastic foundation depends on whether rigid or dead loading is being used; see [31]. Secondly, it is justi ed to use projection boundary conditions for the numerical computations even though they do not correspond to mechanical conditions such as requiring that the shell is simply supported. These results con rm conclusions drawn in [23, 24, 25] in which experimental results are compared to solitary-wave solutions. 10. Conclusions and Discussion. We have investigated the existence and uniqueness of solutions to truncated boundary-value problems for a quite abstract class of equations that includes parabolic equations, see (1.11), as well as elliptic equations such as (10.1) below. Such problems arise frequently in the numerical computation of solitary waves to elliptic systems on in nite cylindrical domains. To

32


summarize our results, and to keep the presentation as simple as possible, we focus on the system (10.1) Duxx + y u + cux + g(; y; u; ry u) = 0;

(x; y) 2 R ; u 2 Rm

with appropriate boundary conditions on ; we emphasize that our results are applicable to a far broader class of systems. Let h(x; y) be a solution to (10.1) for = 0 that satis es lim h(x; y) = p (y)

x!1

uniformly for y 2 . To compute h(x; y) numerically, we replaced the in nite cylindrical domain R by the truncated domain (T? ; T+) , i.e., we consider

Duxx + y u + cux + g(; y; u; ry u) = 0; (x; y) 2 (T? ; T+ ) ; where T? < 0 < T+ for large jT j. We also supply boundary conditions at the left and right faces fTg of the truncated domain. The main issue studied in this

(10.2)

paper is to determine which boundary conditions are admissible, i.e. lead to a wellposed problem. Before discussing this issue, we note that, since (10.1) is invariant under translations in x, we should add a phase condition that breaks the translational symmetry and selects a particular solitary wave from the family generated by the xtranslates of h(x; y). A typical phase condition for u(x; y) is

(10.3)

Z

(T? ;T+ )

h(hx ; hxx)(x; y); (u; ux )(x; y) ? (h; hx )(x; y)iR2m dx dy = 0

where h(x; y) can be replaced by a good initial approximation. First, we consider the case that p+ = p? . In this situation, we proved that periodic boundary conditions (10.4)

u(T?; y) = u(T+; y);

ux (T? ; y) = ux (T+; y)

are always admissible: (10.2) and (10.4) plus a phase condition admit a unique xperiodic solution near u = h(x; y) and = 0 for every large interval T = (T? ; T+ ). Separated boundary conditions such as Dirichlet or Neumann conditions are admissible provided the elliptic system has an additional re ection symmetry x 7! ?x; for (10.1), this means that we seek solutions with c = 0. In general, we have to check the following condition that decides upon admissibility. We cast (10.1) as a dynamical system in the x-variable (10.5)

u x vx =

v

?D?1(y u + cv + g(; y; u; ry u))

and linearize about the asymptotic state p0 (y) := p (y) (10.6)

A :=

0 id ?D?1 (y + gu (0; y; p0; ry p0 )) ?cD?1

at = 0. We consider the operator A that appears in (10.6) on some appropriate function space X of y-dependent functions and calculate the spectrum of A. We assumed that A has no spectrum on the imaginary axis, and can therefore de ne the


33

stable and unstable eigenspaces E s and E u of A that are associated with the parts of the spectrum that lie to the left and right, respectively, of the imaginary axis in the complex plane. Separated boundary conditions can then be expressed as (u(T? ; ); ux (T?; )) 2 E?bc ;

(u(T+ ; ); ux(T+ ; )) 2 E+bc

where Ebc X are certain subspaces of the function space X that encode the boundary conditions: Ebc contain all elements (u; v) 2 X that satisfy the boundary conditions at x = T . For instance, Dirichlet and Neumann conditions correspond to

EDir = f(0; v) 2 X g;

ENeu = f(u; 0) 2 X g;

respectively. The condition that guarantees admissibility is

E?bc E s = X = E+bc E u :

(10.7)

It is a transversality condition of the boundary conditions with the stable and unstable subspaces de ned above. In particular, the so-called projection boundary conditions

E?bc := E u ;

E+bc := E s

always satisfy (10.7). Next, consider the case that p? 6= p+ . Periodic boundary conditions are then meaningless and will not work. Separated boundary conditions are admissible provided

E?bc E?s = X = E+bc E+u

(10.8)

where Es;u are the stable and unstable eigenspaces associated with the linearizations

0 id ?D?1(y + gu (0; y; p; ry p )) ?cD?1

about (p ; 0) at = 0, respectively. The system (10.1) describes, for instance, travelling waves h(x; y) = h( ? ct; t) of the reaction-diusion system (10.9)

ut = Du + y u + g(y; u; u ; ry u);

(; y) 2 R :

In this context, it is of interest to relate the spectrum of the linearization of (10.9) about the travelling wave h to the spectrum of the linearization of the truncated problem (10.2). That involves to determine the fate of isolated eigenvalues with nite multiplicity and of the essential spectrum under truncation. The persistence of isolated eigenvalues can be investigated using the methods presented in this article. Again, an admissibility condition similar to (10.7) or (10.8) is needed. For PDEs on R, some results have recently been obtained in [5]. The break-up of the essential spectrum and the generation of additional eigenvalues due to the presence of boundary conditions is discussed in [32]. Acknowledgments. BS is grateful to Andrew Poje for helpful discussions on discretization issues and for allowing us to use his conjugated-gradient solver.

34

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