Title: Travelling breathers with exponentially small tails in nonlinear oscillator chains

Speaker: G James

Abstract: Travelling breathers are localized oscillations in nonlinear lattices which appear time periodic in a system of reference moving at constant velocity. For Fermi-Pasta-Ulam (FPU) lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgaessner in the case of travelling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact travelling breather solutions superposed on an exponentially small periodic tail. The case of Klein-Gordon lattices is also briefly discussed.

NLDD05, Nonlinear excitations: theory and experiments, Sevilla, March 3-4, 2005.