Asymptotic stability of solitary waves for the 1D cubic-quintic Schr\"odinger equation with no internal mode

TL;DR

This paper proves the asymptotic stability of solitary waves in a 1D cubic-quintic Schrödinger equation, showing stability for many frequencies without internal modes or resonances, advancing understanding of wave stability.

Contribution

It establishes asymptotic stability of solitary waves in a 1D cubic-quintic Schrödinger equation without internal modes, for a broad frequency range.

Findings

Solitary waves are asymptotically stable for many frequencies.

The linearized problem lacks internal modes and resonances.

The stability result applies to a large class of solutions.

Abstract

For the Schr\"odinger equation with a cubic-quintic, focusing-defocusing nonlinearity in one space dimension, we prove the asymptotic stability of solitary waves for a large range of admissible frequencies. For this model, the linearized problem around the solitary waves does not have internal mode nor resonance.