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 $i \, \partial_t ψ+ \partial_x^2 ψ+ |ψ|^2 ψ- g( |ψ|^2 ) ψ= 0$. 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.