Heteroclinic travelling waves in convex FPU-type chains

TL;DR

This paper proves the existence of heteroclinic travelling wave fronts in convex FPU-type chains for a broad class of asymptotic states, extending previous bifurcation results and linking discrete chains to continuum shock solutions.

Contribution

It establishes the existence of heteroclinic fronts for any asymptotic states satisfying certain constraints, generalizing prior bifurcation results and connecting discrete chains to continuum shock profiles.

Findings

Existence of heteroclinic fronts for a wide range of asymptotic states.

Front profiles correspond to energy-conserving supersonic shocks in the continuum limit.

Numerical methods for computing and analyzing front solutions.

Abstract

We consider infinite FPU-type atomic chains with general convex potentials and study the existence of monotone fronts that are heteroclinic travelling waves connecting constant asymptotic states. Iooss showed that small amplitude fronts bifurcate from convex-concave turning points of the force. In this paper we prove that fronts exist for any asymptotic states that satisfy certain constraints. For potentials whose derivative has exactly one turning point these constraints precisely mean that the front corresponds to an energy conserving supersonic shock of the `p-system', which is the naive hyperbolic continuum limit of the chain. The proof goes via minimizing an action functional for the deviation from this discontinuous shock profile. We also discuss qualitative properties and the numerical computation of fronts.