Heteroclinic travelling waves in convex FPU-type chains

We consider infinite FPU-type atomic chains with general convex potentials and study the existence of monotone fronts that are heteroclinic travelling waves connecting constant asymptotic states. Iooss showed that small amplitude fronts bifurcate from convex-concave turning points of the force. In this paper we prove that fronts exist for any asymptotic states that satisfy certain constraints. For potentials whose derivative has exactly one turning point these constraints precisely mean that the front corresponds to an energy conserving supersonic shock of the `p-system', which is the naive hyperbolic continuum limit of the chain. The proof goes via minimizing an action functional for the deviation from this discontinuous shock profile. We also discuss qualitative properties and the numerical computation of fronts.