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.