Travelling waves for a Frenkel-Kontorova chain

TL;DR

This paper proves the existence of heteroclinic traveling wave solutions in a Frenkel-Kontorova chain with smooth and piecewise quadratic on-site potentials, using a Schauder fixed point approach.

Contribution

It introduces a general method to establish heteroclinic waves in Frenkel-Kontorova models with smooth and piecewise quadratic potentials.

Findings

Existence of heteroclinic traveling waves proven.

Method applies to smooth ($C^2$) on-site potentials.

Two-transition waves shown for piecewise quadratic potential.

Abstract

In this article, the Frenkel-Kontorova model for dislocation dynamics is considered, where the on-site potential consists of quadratic wells joined by small arcs, which can be spinodal (concave) as commonly assumed in physics. The existence of heteroclinic waves ---making a transition from one well of the on-site potential to another--- is proved by means of a Schauder fixed point argument. The setting developed here is general enough to treat such a Frenkel-Kontorova chain with smooth ($C^2$) on-site potential. It is shown that the method can also establish the existence of two-transition waves for a piecewise quadratic on-site potential.