Orbital stability: analysis meets geometry

S. de Bievre 1, 2 François Genoud 3 Simona Rota Nodari 2
1 MEPHYSTO - Quantitative methods for stochastic models in physics
Inria Lille - Nord Europe, ULB - Université Libre de Bruxelles [Bruxelles], LPP - Laboratoire Paul Painlevé - UMR 8524
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 Schrödinger equation, for the wave equation, and for the Manakov system.
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S. de Bievre, 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⟩

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