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We consider the Sasa-Satsuma (SS) and Nonlinear Schr\"odinger (NLS) equationsposed along the line, in 1+1 dimensions. Both equations are canonicalintegrable $U(1)$ models, with solitons, multi-solitons and breather solutions,see Yang for instance. For these two equations, we recognize four distinctlocalized breather modes: the Sasa-Satsuma for SS, and for NLS theSatsuma-Yajima, Kuznetsov-Ma and Peregrine breathers. Very little is knownabout the stability of these solutions, mainly because of their complexstructure, which does not fit into the classical soliton behavior byGrillakis-Shatah-Strauss. In this paper we find the natural $H^2$ variationalcharacterization for each of them, and prove that Sasa-Satsuma breathers are$H^2$ nonlinearly stable, improving the linear stability property previouslyproved by Pelinovsky and Yang. Moreover, in the SS case, we provide analternative understanding of the SS solution as a breather, and not only as anembedded soliton. The method of proof is based in the use of a $H^2$ basedLyapunov functional, in the spirit of the first and third authors, extendedthis time to the vector-valued case. We also provide another rigorousjustification of the instability of the remaining three nonlinear modes(Satsuma-Yajima, Peregrine y Kuznetsov-Ma), based in the study of theircorresponding linear variational structure (as critical points of a suitableLyapunov functional), and complementing the instability results recently provede.g. in a paper by the third author.

Stability and instability of breathers in the $U(1)$ Sasa-Satusuma and Nonlinear Schr\"odinger models