On scattering of solitons for wave equation coupled to a particle

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 of the free wave equation. It is assumed that the charge density satisfies the Wiener condition which is a version of the Fermi Golden Rule and that the total charge of the particle equals zero. The proof is based on a development of the general strategy introduced in the papers of Soffer and Weinstein, Buslaev and Perel'man, and others: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.