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

Cite this

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title = "Travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potential",
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.",
keywords = "Anharmonic wells, Frenkel–Kontorova model, Heteroclinic travelling waves, Schauder fixed point theorem",
author = "Boris Buffoni and Hartmut Schwetlick and Johannes Zimmer",
year = "2019",
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day = "1",
doi = "10.1016/j.matpur.2019.01.002",
language = "English",
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pages = "1--40",
journal = "Journal de Mathematiques Pures et Appliquees",
issn = "0021-7824",
publisher = "Elsevier",

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TY - JOUR

T1 - Travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potential

AU - Buffoni, Boris

AU - Schwetlick, Hartmut

AU - Zimmer, Johannes

PY - 2019/3/1

Y1 - 2019/3/1

N2 - 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.

AB - 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.

KW - Anharmonic wells

KW - Frenkel–Kontorova model

KW - Heteroclinic travelling waves

KW - Schauder fixed point theorem

U2 - 10.1016/j.matpur.2019.01.002

DO - 10.1016/j.matpur.2019.01.002

M3 - Article

VL - 123

SP - 1

EP - 40

JO - Journal de Mathematiques Pures et Appliquees

JF - Journal de Mathematiques Pures et Appliquees

SN - 0021-7824

ER -