On Scattering of Solitons for the Klein–Gordon Equation Coupled to a Particle

We establish the long time soliton asymptotics for the translation invariant nonlinear system consisting of the Klein–Gordon equation coupled to a charged relativistic particle. The coupled system has a six dimensional invariant manifold of the soliton solutions. We show that in the large time approximation any finite energy solution, with the initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution of the free Klein–Gordon equation. It is assumed that the charge density satisfies the Wiener condition which is a version of the “Fermi Golden Rule”. The proof is based on an extension of the general strategy introduced by Soffer and Weinstein, Buslaev and Perelman, 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. Supported partly by Austrian Science Foundation (FWF) Project P19138-N13, by research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002). On leave Department Mechanics and Mathematics of Moscow State University. Supported partly by Austrian Science Foundation (FWF) Project P19138-N13 by Max-Planck Institute of Mathematics in the Sciences (Leipzig), and Wolfgang Pauli Institute of Vienna University. Supported partially by the NSF grant DMS-0405927