We consider the Sasa–Satsuma (SS) and nonlinear Schrdinger (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 H ² 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 H ² 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 H ² based Lyapunov functional, in the spirit of Alejo and Muoz (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 Muoz (2017 Proyecciones (Antofagasta) 36 653–83).

我们考虑沿线提出的 Sasa-Satsuma (SS) 和非线性 Schrdinger (NLS) 方程，在 1 + 1 维中。这两个方程都是典型的可积U (1) 模型，具有孤子、多孤子和呼吸解杨（2010 SIAM 数学建模和计算）。对于这两个方程，我们识别出四种不同的局部呼吸模式：SS 的 Sasa-Satsuma，以及 NLS 的 Satsuma-Yajima、Kuznetsov-Ma 和 Peregrine 呼吸模式。人们对这些解的稳定性知之甚少，主要是因为它们的结构复杂，不符合经典孤子行为 Grillakis等人(1987 J. Funct. Anal. 74 160–97)。在本文中，我们发现自然H ²他们每个人的变分表征。这似乎是这些解的第一个已知变分表征；特别是第一个为著名的 Peregrine 呼吸器而获得的。我们还证明了 Sasa-Satsuma 呼吸器是H ²非线性稳定的，改善了先前由 Pelinovsky 和 ​​Yang (2005 Chaos 15 037115)证明的线性稳定性特性。此外，在 SS 的情况下，我们将 SS 解决方案作为呼吸器提供了另一种理解，而不仅仅是作为嵌入式孤子。证明方法基于使用基于H ²的李雅普诺夫泛函，本着 Alejo 和 Muoz (2013 Commun. Math. Phys. 324233-62），这次扩展到向量值的情况。我们还提供了其余三种非线性模式（Satsuma-Yajima、Peregrine 和 Kuznetsov-Ma）的不稳定性的另一个严格证明，基于对它们相应的线性变分结构（作为合适的 Lyapunov 函数的临界点）的研究，并补充最近在Muoz ( 2017 Proyecciones ( Antofagasta ) 36 653–83) 中证明了不稳定性结果。