  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 ($C^2$) 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.
