HAL : hal-00684236, version 1
 Existence of supersonic traveling waves for the Frenkel-Kontorova model
 Samar Issa 1, Mustapha Jazar 1
 (31/03/2012)
 In this paper, we study the standard one-dimensional (non-overdamped) Frenkel-Kontorova (FK) model describing the motion of atoms in a lattice. For this model we show that for any supersonic velocity $c>1$, there exist bounded traveling waves moving with velocity $c$. The profile of these traveling waves is a phase transition between limit states $k_-$ in $-\infty$ and $k_+$ in $+\infty$. Those limit states are some integers which reflect the assumed $1$-periodicity of the periodic potential inside the FK model. For every $c>1$, we show that we can always find $k_-$ and $k_+$ such that $k_+-k_-$ is an odd integer. Furthermore for $c\ge \sqrt{\frac{25}{24}}$, we show that we can take $k_+-k_-=1$. These traveling waves are limits of minimizers of a certain energy functional defined on a bounded interval, when the length of the interval goes to infinity. Our method of proof uses a concentration compactness type argument which is based on a cleaning lemma for minimizers of this functional.
 1 : Mathematics Department Lebanese University 2 : Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS) Ecole des Ponts ParisTech
 Domaine : Mathématiques/Equations aux dérivées partielles
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 hal-00684236, version 1 http://hal.archives-ouvertes.fr/hal-00684236 oai:hal.archives-ouvertes.fr:hal-00684236 Contributeur : Régis Monneau <> Soumis le : Samedi 31 Mars 2012, 07:31:29 Dernière modification le : Dimanche 1 Avril 2012, 16:04:22