Orbital stability: analysis meets geometry
Analysis of PDEs
2015-01-07 v2 Dynamical Systems
Symplectic Geometry
Abstract
We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system.
Cite
@article{arxiv.1407.5951,
title = {Orbital stability: analysis meets geometry},
author = {Stephan De Bievre and François Genoud and Simona Rota Nodari},
journal= {arXiv preprint arXiv:1407.5951},
year = {2015}
}