Scattering of Solitons for Coupled Wave-Particle Equations

TL;DR

This paper proves that solutions near a soliton manifold in a coupled wave-particle system asymptotically decompose into a soliton and a dispersive wave, under specific conditions on charge density and initial momenta.

Contribution

It establishes long-time asymptotics for coupled wave-particle equations with a six-dimensional soliton manifold, extending previous methods to this complex system.

Findings

Solutions decompose into soliton plus dispersive wave over time

Conditions on charge density ensure stability and decay

Method adapts symplectic projection and modulation techniques

Abstract

We establish a long time soliton asymptotics for a nonlinear system of wave equation coupled to a charged particle. The coupled system has a six dimensional manifold of soliton solutions. We show that in the large time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution to the free wave equation. It is assumed that the charge density satisfies Wiener condition which is a version of Fermi Golden Rule, and that the momenta of the charge distribution vanish up to the fourth order. The proof is based on a development of the general strategy introduced by Buslaev and Perelman: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.