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Scattering of solitons for coupled wave-particle equations. (English) Zbl 1235.35068

Summary: 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 to the free wave equation. It is assumed that the charge density satisfies Wiener’s condition, which is a version of Fermi’s golden rule, and that the momenta of the charge distribution vanish up to the fourth order. The proof is based on a development of the general strategy introduced by V. S. Buslaev and G. Perelman [in: Méthodes semi-classiques, Volume 2. Colloque international (Nantes, juin 1991). Paris: Société Mathématique de France, Astérisque. 210, 49–63 (1992; Zbl 0795.35111)]: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.

MSC:

35C08 Soliton solutions
35L10 Second-order hyperbolic equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q40 PDEs in connection with quantum mechanics

Citations:

Zbl 0795.35111
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References:

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