Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schr\"odinger equation

TL;DR

This paper proves the local asymptotic stability of small standing solitary waves in a one-dimensional cubic-quintic Schrödinger equation, demonstrating their robustness under even perturbations in the energy space.

Contribution

It establishes the asymptotic completeness of small solitary waves for the cubic-quintic Schrödinger equation, incorporating analysis of internal modes via the Fermi golden rule.

Findings

Small solitary waves are asymptotically stable under even perturbations.

Internal modes are effectively controlled using the Fermi golden rule.

The results extend stability analysis to a non-integrable, focusing-focusing nonlinear Schrödinger model.

Abstract

For the Schr\"odinger equation with a cubic-quintic, focusing-focusing nonlinearity in one space dimension, this article proves the local asymptotic completeness of the family of small standing solitary waves under even perturbations in the energy space. For this model, perturbative of the integrable cubic Schr\"odinger equation for small solutions, the linearized equation around a small solitary wave has an internal mode, whose contribution to the dynamics is handled by the Fermi golden rule.