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On scattering of solitons for the Klein-Gordon equation coupled to a particle

V. Imaikin·A. Komech·B. Vainberg
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Abstract

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.

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V. Imaikin, A. Komech, B. Vainberg. 2006-09-08. On scattering of solitons for the Klein-Gordon equation coupled to a particle. https://doi.org/10.1007/s00220-006-0088-z

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