Existence of supersonic traveling waves for the Frenkel-Kontorova model

Samar Issa (1) , Mustapha Jazar (1) , Régis Monneau (2)

1 Mathematics Department
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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 -infinity and k(+) in +infinity. 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 >= root 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.

Domaines: Equations aux dérivées partielles [math.AP]

Samar Issa, Mustapha Jazar, Régis Monneau. Existence of supersonic traveling waves for the Frenkel-Kontorova model. Differential and integral equations, 2013, 26 (3-4), pp.321-353. ⟨hal-00684236⟩