Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schr\"odinger equation in the absence of internal modes

TL;DR

This paper proves the asymptotic stability of solitary waves in a 1D near-cubic nonlinear Schrödinger equation by showing the absence of internal modes and resonances in the linearized problem, under certain conditions.

Contribution

It establishes the asymptotic stability of solitary waves for a class of perturbed cubic NLS equations without internal modes or resonances.

Findings

Linearized problem around solitary waves has no internal modes

Solitary waves are asymptotically stable for small frequencies

Conditions on the function g ensure stability and absence of resonances

Abstract

We consider perturbations of the one-dimensional cubic Schr\"odinger 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.