## On dispersion of small energy solutions of the nonlinear Klein-Gordon equation with a potential. (On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential.)(English)Zbl 1237.35115

Authors’ abstract: We study small amplitude solutions of nonlinear Klein-Gordon equations with a potential. Under smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity, we prove that all small amplitude initial data with finite energy give rise to solutions asymptotically free. In the case where the linear system has at most one bound state the result was already proved by A. Soffer and M. I. Weinstein [Invent. Math. 136, No. 1, 9–74 (1999; Zbl 0910.35107)]; we obtain here a result valid in the case of an arbitrary number of possibly degenerate bound states. The proof is based on a combination of Birkhoff normal form techniques and dispersive estimates.

### MSC:

 35L71 Second-order semilinear hyperbolic equations 35Q55 NLS equations (nonlinear Schrödinger equations) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35B40 Asymptotic behavior of solutions to PDEs 37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Zbl 0910.35107
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