Nonlinear Scattering in the Lamb System

Abstract

The goal of this paper is to survey recent results on scattering in nonlinear conservative Lamb’ s systems. A Lamb’ s system is a wave equation coupled with an equation of motion of a particle of mass \(m\geq 0.\)We describe the long time asymptotics in a global energy norm for all finite energy solutions with \(m=0[6]\) and \(m> 0[7].\)Under certain conditions on the potential, each solution in an appropriate functional space decays, in a global energy norm asc \(t\rightarrow\pm \infty,\)towards the sum of a stationary state and an outgoing wave. The outgoing waves correspond to the ’in’ and ’out’ asymptotic states. For \(m > 0,\)we define nonlinear wave operators corresponding to the ones introduced in [6] and obtain a necessary condition for the existence of the asymptotic states. For m = 0 we state a conjecture for the asymptotic completeness and verify this for some particular potentials.