Existence of supersonic traveling waves for the Frenkel-Kontorova model

Abstract

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 $-\infty$ and $k_+$ in $+\infty$. 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 $c\ge \sqrt{\frac{25}{24}}$, 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.