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 $i{\partial }_t\psi +{\partial }_x^2\psi +|\psi {|}^2\psi -g(|\psi {|}^2)\psi =0$. 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.

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