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

Authors
  • De Bièvre, Stephan1, 2
  • Rota Nodari, Simona3
  • 1 Université de Lille, Laboratoire Paul Painlevé, CNRS, UMR 8524 et UFR de Mathématiques, Villeneuve d’Ascq Cedex, 59655, France , Villeneuve d’Ascq Cedex (France)
  • 2 Centre de Recherche INRIA Futurs, Parc Scientifique de la Haute Borne, Equipe-Projet MEPHYSTO, 40, Avenue Halley, Villeneuve d’Ascq Cedex, 59658, France , Villeneuve d’Ascq Cedex (France)
  • 3 Université Bourgogne Franche-Comté, Institut de Mathématiques de Bourgogne (IMB), CNRS, UMR 5584, Dijon, 21000, France , Dijon (France)
Type
Published Article
Journal
Archive for Rational Mechanics and Analysis
Publisher
Springer Berlin Heidelberg
Publication Date
Jul 07, 2018
Volume
231
Issue
1
Pages
233–284
Identifiers
DOI: 10.1007/s00205-018-1278-5
Source
Springer Nature
License
Yellow

Abstract

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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