Review on the stability of the Peregrine and related breathers

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.