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On scattering of solitons for the Klein-Gordon equation coupled to a particle. (English) Zbl 1127.35054

Summary: 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.

MSC:

35Q40 PDEs in connection with quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35P25 Scattering theory for PDEs
35Q51 Soliton equations
35L70 Second-order nonlinear hyperbolic equations
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