Orbital stability: analysis meets geometry

Stephan De Bièvre (1, 2) , François Genoud (3) , Simona Rota Nodari (2)

1 MEPHYSTO - Quantitative methods for stochastic models in physics
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 Faculty of Mathematics [Vienna]

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 Schrödinger equation, for the wave equation, and for the Manakov system.

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie symplectique [math.SG]; Systèmes dynamiques [math.DS]

Stephan De Bièvre, François Genoud, Simona Rota Nodari. Orbital stability: analysis meets geometry. Nonlinear Optical and Atomic Systems, 2146, pp.147-273, 2015, Lecture Notes in Mathematics, ⟨10.1007/978-3-319-19015-0_3⟩. ⟨hal-01028168v3⟩