Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 233–284 | Cite as

Orbital Stability via the Energy–Momentum Method: The Case of Higher Dimensional Symmetry Groups

  • Stephan De BièvreEmail author
  • Simona Rota Nodari


We consider the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We prove a persistence result for such relative equilibria, present a generalization of the Vakhitov–Kolokolov slope condition to this higher dimensional setting, and show how it allows one to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of relative equilibria of nonlinear Schrödinger and Manakov equations. We provide a comparison of our approach to the one by Grillakis–Shatah–Strauss.


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This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and by FEDER (PIA-LABEX-CEMPI 42527). The authors are grateful to Dr. M. Conforti, Prof. F. Genoud, Prof. S. Keraani, Prof. S. Mehdi, Prof S. Trillo and Prof. G. Tuynman for helpful discussions on the subject matter of this paper. They also thank an anonymous referee for instructive comments and a careful reading of the manuscript.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire Paul Painlevé, CNRS, UMR 8524 et UFR de MathématiquesUniversité de LilleVilleneuve d’Ascq CedexFrance
  2. 2.Equipe-Projet MEPHYSTOCentre de Recherche INRIA Futurs, Parc Scientifique de la Haute BorneVilleneuve d’Ascq CedexFrance
  3. 3.Institut de Mathématiques de Bourgogne (IMB), CNRS, UMR 5584Université Bourgogne Franche-ComtéDijonFrance

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