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We consider the Sasa–Satsuma (SS) and nonlinear Schrödinger (NLS) equations posed along the line, in 1 + 1 dimensions. Both equations are canonical integrable U(1) models, with solitons, multi-solitons and breather solutions Yang (2010 SIAM Mathematical Modeling and Computation). For these two equations, we recognize four distinct localized breather modes: the Sasa–Satsuma for SS, and for NLS the Satsuma–Yajima, Kuznetsov–Ma and Peregrine breathers. Very little is known about the stability of these solutions, mainly because of their complex structure, which does not fit into the classical soliton behavior Grillakis et al (1987 J. Funct. Anal. 74 160–97). In this paper we find the natural H2 variational characterization for each of them. This seems to be the first known variational characterization for these solutions; in particular, the first one obtained for the famous Peregrine breather. We also prove that Sasa–Satsuma breathers are H2 nonlinearly stable, improving the linear stability property previously proved by Pelinovsky and Yang (2005 Chaos 15 037115). Moreover, in the SS case, we provide an alternative understanding of the SS solution as a breather, and not only as an embedded soliton. The method of proof is based in the use of a H2 based Lyapunov functional, in the spirit of Alejo and Muńoz (2013 Commun. Math. Phys. 324 233–62), extended this time to the vector-valued case. We also provide another rigorous justification of the instability of the remaining three nonlinear modes (Satsuma–Yajima, Peregrine and Kuznetsov–Ma), based in the study of their corresponding linear variational structure (as critical points of a suitable Lyapunov functional), and complementing the instability results recently proved e.g. in Muńoz (2017 Proyecciones (Antofagasta) 36 653–83). [ABSTRACT FROM AUTHOR]

This paper aims to provide a robust model for the well-known phenomenon of Kármán vortex street arising in fluid mechanics. The first theoretical attempt to model this pattern was given by von Kármán (1991 Göttinger Nachr., Math. Phys. Kl. 509–17); von Kármán (1912 Göttinger Nachr., Math. Phys. Kl. 547–56). He considered two parallel staggered rows of point vortices, with opposite strength, that translate at the same speed. Following the ideas of Saffman and Schatzman (1981 SIAM J. Sci. Stat. Comput. 2 285–95), we propose to study this phenomenon in the Euler equations by considering two infinite arrows of vortex patches. The key idea is to desingularize the point vortex model proposed by von Kármán. Our construction is flexible and can be extended to more general incompressible models. [ABSTRACT FROM AUTHOR]