Review on the stability of the Peregrine and related breathers

TL;DR

This paper reviews the stability properties of Peregrine, Kuznetsov-Ma, and Akhmediev breathers in nonlinear Schrödinger equations, showing they are unstable in energy spaces and discussing related spectral and Lyapunov analyses.

Contribution

It provides a comprehensive stability analysis of key NLS breathers, introducing Lyapunov functionals and spectral properties, and discusses the concept of global stability and open questions.

Findings

All three breathers are unstable in energy spaces.

Lyapunov functionals and spectral properties are characterized.

Nonlinear ODEs for the breathers are derived.

Abstract

In this note, we review stability properties in energy spaces of three important nonlinear Schr\"odinger breathers: Peregrine, Kuznetsov-Ma, and Akhmediev. More precisely, we show that these breathers are unstable according to a standard definition of stability. Suitable Lyapunov functionals are described, as well as their underlying spectral properties. As an immediate consequence of the first variation of these functionals, we also present the corresponding nonlinear ODEs fulfilled by these NLS breathers. The notion of global stability for each breather above mentioned is finally discussed. Some open questions are also briefly mentioned.