In this paper we study small amplitude solutions of nonlinear Klein Gordon equations with a potential. Under suitable smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity, we prove that all small energy solutions are asymptotically free. In cases where the linear system has at most one bound state the result was already proved by Soffer and Weinstein: 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.

We use the inverse scattering transform, the auto-Bäcklund transformation, and the steepest descent method of Deift and Zhou to obtain the asymptotic stability of the solitons in the cubic NLS (nonlinear Schrödinger) equation.