Existence of supersonic traveling waves for the Frenkel-Kontorova model

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 −∞ and k+ in +∞. 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 ≥ √ 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. AMS Classification: 35A15, 35C07, 37K60.