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 ≥ 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 as → ±∞, 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.