A review on asymptotic stability of solitary waves in nonlinear dispersive problems in dimension one

TL;DR

This paper reviews the current understanding of the asymptotic stability of solitary waves in one-dimensional nonlinear dispersive equations, emphasizing spectral methods and broader contexts like Klein-Gordon equations.

Contribution

It provides a comprehensive overview of the state of the art in asymptotic stability analysis for solitary waves, highlighting spectral techniques and recent developments.

Findings

Spectral methods are central to stability analysis.

Full asymptotic stability involves decomposition into solitary wave and decaying part.

The review covers nonlinear Schrödinger and Klein-Gordon equations.

Abstract

We review asymptotic stability of solitary waves for nonlinear dispersive equations set on the line. Our focus is threefold: first, the nonlinear Schrodinger equation; second, the notion of full asymptotic stability (which states that perturbations of a solitary wave decompose globally into a solitary wave and a decaying solution); and third, spectral methods. Besides this focus, we summarize the state of the art in a broader context, including nonlinear Klein-Gordon equations, the notion of local asymptotic stability, and virial methods.