Traveling wave solutions for the accelerated Frenkel-Kontorova model: The monostable cases

Abstract

In this paper, we consider a system of accelerated and general fully non-linear discrete equations depending on a parameter σ lying inside an interval $[{\sigma }^{-},{\sigma }^{+}]$. For $\sigma \in ({\sigma }^{-},{\sigma }^{+})$, our non-linearity is bistable and for $\sigma ={\sigma }^{\pm }$, it is monostable. Two results are obtained: the first one is to derive properties of the velocity function associated to the existence of traveling waves in the bistable regimes. The second one is to construct traveling waves in the monostable regimes. Our approach is to consider the monostable regimes as the limit of bistable ones. As far as we know, this is the first result concerning traveling waves for accelerated, general and monostable fully-nonlinear discrete system.