Travelling waves in the Fermi-Pasta-Ulam lattice

We consider travelling wave solutions on a one-dimensional lattice, corresp onding to mass particles interacting nonlinearly with their nearest neighbo ur (the Fermi-Pasta-Ulam model). A constructive method is given, for obtain ing all small bounded travelling waves for generic potentials, near the fir st critical value of the velocity. They all are given by solutions of a fin ite-dimensional reversible ordinary differential equation. In particular, n ear (above) the first critical velocity of the waves, we construct the soli tary waves (localized waves with the basic state at infinity) whose global existence was proved by Friesecke and Wattis, using a variational approach. In addition, we find other travelling waves such as (a) a superposition of a periodic oscillation with a non-zero uniform stretching or compression b etween particles, (b) mainly localized waves which tend towards a uniformly stretched or compressed lattice at infinity, (c) heteroclinic solutions co nnecting a stretched pattern with a compressed one.