Travelling waves for a Frenkel-Kontorova chain

In this article, the Frenkel-Kontorova model for dislocation dynamics is considered, where the on-site potential consists of quadratic wells joined by small arcs, which can be spinodal (concave) as commonly assumed in physics. The existence of heteroclinic waves-making a transition from one well of the on-site potential to another-is proved by means of a Schauder fixed point argument. The setting developed here is general enough to treat such a Frenkel-Kontorova chain with smooth (C2) on-site potential. It is shown that the method can also establish the existence of two-transition waves for a piecewise quadratic on-site potential.