Asymptotic Stability and Completeness in the Energy Space for Nonlinear Schrödinger Equations with Small Solitary Waves

Abstract

In this paper we study a class of nonlinear Schrödinger equations which admit families of small solitary wave solutions. We consider solutions which are small in the energy space $H^1$, and decompose them into solitary wave and dispersive wave components. The goal is to establish the asymptotic stability of the solitary wave and the asymptotic completeness of the dispersive wave. That is, we show that as $t \to \infty$, the solitary wave component converges to a fixed solitary wave, and the dispersive component converges to a solution of the free Schrödinger equation.