Birkhoff Normal Form and Hamiltonian PDEs

By Benoît Grébert. 2006

91 citations recorded by the source corpus.

These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centered on the Birkhoff normal form theorem that we first proved in finite dimension. Then, after giving some exemples of Hamiltonian PDEs, we present an abstract Birkhoff normal form theorem in infinite dimension and discuss the dynamical consequences for Hamiltonian PDEs.

Subfields: Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/563c657898b66addfd8da8f6b7cbb690b98c5a8d

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In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation (2010, methodology): “There are many related results, extending the analysis too to higher dimensional partial differential equations, see e.g. the works of Bambusi [3], Grebert [47], Eliasson-Kuksin [45], Bourgain [23], and references therein.”
Resonant dynamics for the quintic nonlinear Schrödinger equation (2011, unknown): “Bambusi and the first author [2] (see also [6]) that the linear actions |ξj |2, j ∈ Z, are almost invariant during very long time, or more precisely, that for all N 1 ∣∣ξj (t)∣∣2 = ∣∣ξj (0)∣∣2 +O(ν), for |t | ν−N.”
A Birkhoff normal form theorem for some semilinear PDEs (2008, methodology): “The proof of this first case is (up to minor changes) equal to that given in [ Gre06 ].”