Existence and Uniqueness of Traveling Waves for Fully Overdamped Frenkel–Kontorova Models

Abstract

In this article, we study the existence and the uniqueness of traveling waves for a discrete reaction–diffusion equation with bistable nonlinearity, namely a generalization of the fully overdamped Frenkel–Kontorova model. This model consists of a system of ODEs which describes the dynamics of crystal defects in lattice solids. Under very weak 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.