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

Boris Buffoni, Hartmut Schwetlick, Johannes Zimmer

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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 propose a novel approach: 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.
Original languageEnglish
Pages (from-to)1-40
Number of pages40
JournalJournal de Mathématiques Pures et Appliquées
Volume123
Early online date22 Jan 2019
DOIs
Publication statusPublished - 1 Mar 2019

Keywords

  • Anharmonic wells
  • Frenkel–Kontorova model
  • Heteroclinic travelling waves
  • Schauder fixed point theorem

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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