Exit manifolds for lattice differential equations

TL;DR

This paper investigates the interaction of well-separated traveling waves in one-dimensional lattice differential equations, proving the existence and stability of solutions resembling superpositions of these waves moving in opposite directions.

Contribution

It introduces a novel approach to analyze interactions of traveling waves in lattice differential equations by embedding the system in a larger framework to handle non-shift-periodic solutions.

Findings

Existence of solutions as superpositions of traveling waves

Such solutions are asymptotically stable

Method to handle non-shift-periodic wave interactions

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 Chow, Mallet-Paret and Shen) traveling wave solutions whose profiles approach limiting equilibria exponentially fast, then the system admits solutions which are nearly the linear superposition of two such traveling 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 traveling waves are moving in opposite directions, these solutions are not shift-periodic and hence the framework of Chow, Mallet-Paret, and Shen 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, even though it is non-autonomous and large, has certain properties which allow us to treat it as if it were a small perturbation.