Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes

Abstract

We consider perturbations of the one-dimensional cubic Schrödinger equation, under the form i∂tψ+∂x2ψ+|ψ|2ψ−g(|ψ|2)ψ=0. Under hypotheses on the function g that can be easily verified in some cases, we show that the linearized problem around a solitary wave does not have internal mode (nor resonance) and we prove the asymptotic stability of these solitary waves, for small frequencies.