We study the long-time behavior of solutions of the nonlinear Schrödinger equation in one space dimension for initial conditions in a small neighborhood of a stable solitary wave. Under some hypothesis on the structure of the spectrum of the linearized operator, we prove that, asymptotically in time, the solution decomposes into a solitary wave with slightly modified parameters and a dispersive part described by the free Schrödinger equation. We explicitly calculate the time behavior of the correction. © 2003 Éditions scientifiques et médicales Elsevier SAS.
Buslaev, V. S., & Sulem, C. (2003). On asymptotic stability of solitary waves for nonlinear schrödinger equations. Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire, 20(3), 419–475. https://doi.org/10.1016/S0294-1449(02)00018-5