Orbital stability via the energy-momentum method: the case of higher dimensional symmetry groups

TL;DR

This paper extends the energy-momentum method to analyze the orbital stability of relative equilibria in Hamiltonian systems with higher-dimensional symmetry groups, applying it to nonlinear Schrödinger and Manakov equations.

Contribution

It generalizes the Vakhitov-Kolokolov slope condition for higher-dimensional invariance groups and demonstrates its effectiveness in establishing orbital stability.

Findings

Proves a persistence result for relative equilibria.

Generalizes the Vakhitov-Kolokolov slope condition.

Shows local coercivity of the Lyapunov function implies stability.

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 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 Schr{\"o}dinger and Manakov equations. We provide a comparison of our approach to the one by Grillakis-Shatah-Strauss.