Travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potential

TL;DR

This paper proves the existence of travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potentials, extending previous harmonic-well results through advanced dynamical systems techniques.

Contribution

It introduces a global centre manifold theory for anharmonic wave trains and constructs heteroclinic travelling waves in a nonlocal, nonlinear, nonconvex setting.

Findings

Existence of heteroclinic travelling waves for anharmonic potentials

Development of a global centre manifold theory for nonconvex on-site potentials

Asymptotic approximation of solutions using centre manifold parametrization

Abstract

The Frenkel-Kontorova model for dislocation dynamics from 1938 is given by a chain of atoms, where neighbouring atoms interact through a linear spring and are exposed to a smooth periodic on-site potential. A dislocation moving with constant speed corresponds to a heteroclinic travelling wave, making a transition from one well of the on-site potential to another. The ensuing system is nonlocal, nonlinear and nonconvex. We present an existence result for a class of smooth nonconvex on-site potentials. Previous results in mathematics and mechanics have been limited to on-site potentials with harmonic wells. To overcome this restriction, we first develop a global centre manifold theory for anharmonic wave trains, then parametrise the centre manifold to obtain asymptotically correct approximations to the solution sought, and finally obtain the heteroclinic wave via a fixed point argument.