Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models

Mohammad Al Haj (1) , Nicolas Forcadel (2, 3) , Régis Monneau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems

In this article, we study the existence and the uniqueness of traveling waves for a discrete reaction-diffusion equation with bistable non-linearity, namely a generalization of the fully overdamped Frenkel-Kontorova model. This model consists in a system of ODE's which describes the dynamics of crystal defects in a lattice solids. Under very poor assumptions, we prove the existence of a traveling wave solution and the uniqueness of the velocity of propagation of this traveling wave. The question of the uniqueness of the profile is also studied by proving Strong Maximum Principle or some weak asymptotics on the profile at infinity.

Mots clés: Frenkel-Kontorova models; traveling waves; viscosity solutions; comparison principle

Domaines: Equations aux dérivées partielles [math.AP]

Mohammad Al Haj, Nicolas Forcadel, Régis Monneau. Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models. 2012. ⟨hal-00721233v1⟩