Orbital Stability: Analysis Meets Geometry

Abstract

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.

Existence and Stability of Traveling Waves for Discrete Nonlinear Schrödinger Equations Over Long Times

“…Then it has been developed, in the general context of Hamiltonian systems by Grillakis, Shatah, Strauss in [10], [11]. A clear and rigorous presentation of the method and its formalism in a general setting is given in the paper [7] of De Bièvre, Genoud, and Rota Nodari.…”

“…There is at least one class of solutions that can be defined without ambiguity, the standing waves (i.e. when ξ 2 " 0) which are solutions of the form (7) @t 0 P R, @t P R, upt 0`t q " e iξ1t uptq.…”

“…This second method is more quantitative than the first one, and the estimates of stability we give in this article are all based on it. It has been developed by Grillakis, Shatah and Strauss in 1987 in [10] and [11] (see also [7] for a very clear presentation of this method). Theorem 1.1.…”

Orbital Stability via the Energy–Momentum Method: The Case of Higher Dimensional Symmetry Groups

“…In [17] and [34], the theory is worked out in a Hilbert space setting, and when the symmetry group G is a one-dimensional Lie group. More recently, in [10], a version of the energy-momentum method has been presented for Hamiltonian dynamical systems on a Banach space E having as invariance group a Lie group G of arbitrary finite dimension. What is shown there is that the proof of orbital stability can be reduced to a "local coercivity estimate" on an appropriately constructed Lyapunov function L. It is shown in [10] that, in the above infinite dimensional setting, the construction of the latter follows naturally from the Hamiltonian structure and basic properties of the momentum map, in complete analogy with the finite dimensional situation.…”

“…More recently, in [10], a version of the energy-momentum method has been presented for Hamiltonian dynamical systems on a Banach space E having as invariance group a Lie group G of arbitrary finite dimension. What is shown there is that the proof of orbital stability can be reduced to a "local coercivity estimate" on an appropriately constructed Lyapunov function L. It is shown in [10] that, in the above infinite dimensional setting, the construction of the latter follows naturally from the Hamiltonian structure and basic properties of the momentum map, in complete analogy with the finite dimensional situation. In specific models, it then remains to show the appropriate local coercivity estimate on L which amounts to a lower bound on its Hessian restricted to an appropriate subspace of E (see (2.14)).…”