Section: New Results

Orbital stability in the presence of symmetries

With S. Rota Nodari, S. De Bièvre considered 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 [9]. They proved a persistence result for such relative equilibria, presented a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and showed how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method was applied to study the orbital stability of relative equilibria of nonlinear Schrödinger and Manakov equations. It extends and clarifies the approach of Grillakis-Shatah-Strauss.