AUTHOR=Alejo Miguel A. , Fanelli Luca , Muñoz Claudio TITLE=Review on the Stability of the Peregrine and Related Breathers JOURNAL=Frontiers in Physics VOLUME=8 YEAR=2020 URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.591995 DOI=10.3389/fphy.2020.591995 ISSN=2296-424X ABSTRACT=

In this note, we review stability properties in energy spaces of three important nonlinear Schrödinger 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 nonlinear Schrödinger breathers. The notion of global stability for each breather mentioned above is finally discussed. Some open questions are also briefly mentioned.