  Traveling waves for the FPU chain are constructed by solving the associated
equation for the spatial profile $u$ of the wave. We consider solutions whose
derivatives $u'$ need not be small, may change sign several times, but decrease
at least exponentially. Our method of proof is computer-assisted. Unlike other
methods, it does not require that the FPU potential has an attractive
(positive) quadratic term. But we currently need to restrict the size of that
term. In particular, our solutions in the attractive case are all supersonic.
