On Scattering of Solitons for Maxwell Equation Coupled to a Particle

We establish a long time soliton asymptotics for a nonlinear system of Maxwell 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 Maxwell equation. It is assumed that the charge density satisfies the Wiener condition. The proof is based on a development of the general strategy introduced in the papers of 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 DFG grant 436 RUS 113/929/0-1, FWF Project P19138-N13, RFBR grant 07-01-00018a, and RFBR-DFG grant 08-01-91950-NNIOa. Supported partly by Alexander von Humboldt Research Award.