Travelling waves in a chain of coupled nonlinear oscillators

In a chain of nonlinear oscillators, linearly coupled to their nearest neig hbors, all travelling waves of small amplitude are found as solutions of fi nite dimensional reversible dynamical systems. The coupling constant and th e inverse wave speed form the parameter space. The groundstate consists of a one-parameter family of periodic waves. It is realized in a certain param eter region containing all cases of light coupling. Beyond the border of th is region the complexity of wave-forms increases via a succession of bifurc ations. In this paper we give an appropriate formulation of this problem, p rove the basic facts about the reduction to finite dimensions, show the exi stence of the ground states and discuss the first bifurcation by determinin g a normal form for the reduced system. Finally we show the existence of na nopterons, which are localized waves with a noncancelling periodic tail at infinity whose amplitude is exponentially small in the bifurcation paramete r.