Exit manifolds for lattice differential equations

Abstract

We study the weak interaction between a pair of well-separated coherent structures in possibly non-local lattice differential equations. In particular, we prove that if a lattice differential equation in one space dimension has asymptotically stable (in the sense of a paper by Chow et al.) travelling-wave solutions whose profiles approach limiting equilibria exponentially fast, then the system admits solutions which are nearly the linear superposition of two such travelling waves moving in opposite directions away from one another. Moreover, such solutions are themselves asymptotically stable. This result is meant to complement analytic or numeric studies into interactions of such pulses over finite times which might result in the scenario treated here. Since the travelling waves are moving in opposite directions, these solutions are not shift-periodic and hence the framework of Chow et al. does not apply. We overcome this difficulty by embedding the original system in a larger one wherein the linear part can be written as a shift-periodic piece plus another piece which, although it is non-autonomous and large, has certain properties which allow us to treat it as if it were a small perturbation.