Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model

Nicolas Forcadel (1) , Amin Ghorbel (2) , Sonda Walha (1, 2)

1 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie
2 FSS - Faculté des Sciences de Sfax

In this paper, we study the existence and uniqueness of traveling wave solution for the accelerated Frenkel-Kontorova model. This model consists in a system of ODE that describes the motion particles in interaction. The most important applications we have in mind is the motion of crystal defects called dislocations. For this model, we prove the existence of traveling wave solutions under very weak assumptions. The uniqueness of the velocity is also studied as well as the uniqueness of the profile which used different types of strong maximum principle. As far as we know, this is the first result concerning traveling waves for accelerated, spatially discrete system.

Keywords: Frenkel-Kontorova models; Traveling waves; Viscosity solutions; Maximum principle; Hull function

Domains: Analysis of PDEs [math.AP]

Nicolas Forcadel, Amin Ghorbel, Sonda Walha. Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model. Journal of Dynamics and Differential Equations, 2014, 26 (4), pp.1133-1169. ⟨hal-00945772⟩