On the Korteweg-de Vries equation: Convergent Birkhoff normal form
type: Item
instance of: scholarly article
title: On the Korteweg-de Vries equation: Convergent Birkhoff normal form
zbMATH Open document ID: 0868.35099
published in: Journal of Functional Analysis
publication date: 1996-10-13T00:00:00Z
review text: The authors consider the Korteweg-de Vries (KdV) equation \[ V_t+ V_{xxx}- 6V\cdot V_x=0 \] on the circle, i.e., for functions \(V=V(x,t)\) on \(S^1\times\mathbb{R}\). It is well-known that the KdV equation represents a completely integrable Hamiltonian system (of infinite dimension) with phase space \(H^n(S^1;\mathbb{R})\), where \(n\) is a positive integer, and with a Poisson structure inducing a symplectic structure on the phase space. The main result of the present paper states that there is a symplectic morphism from the zero-leaf of the phase space, with respect to the Casimir function defined by the Poisson structure to some symplectic Hilbert space. This symplectomorphism is bijective, real-analytic in both directions, and induces globally defined real-analytic action-angle coordinates for the entire KdV hierarchy. This theorem provides the (apparently) first infinite-dimensional counterpart of the result by \textit{J. Vey} [Am. J. Math. 100, 591-614 (1978; Zbl 0384.58012)], which states that a finite-dimensional completely integrable Hamiltonian system with real-analytic Hamiltonian functional admits action-angle variables in a neighborhood of a non-resonant elliptic fixed point. In the second part of the paper, the authors apply their main result to the study of the regularity of the KdV-Hamiltonian vector field. In particular, the local existence of a convergent Birkhoff normal form is proved.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 934260
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Poisson structures
zbMATH Keywords: symplectic spaces
zbMATH Keywords: Birkhoff normal form
author: Daniel Bättig
author: T. Kappeler
author: Boris Mityagin
reviewed by: Werner Kleinert
MaRDI profile type: Publication
full work available at URL: jfan.1996.0111
OpenAlex ID: W1999106139
DOI: 10.1006/JFAN.1996.0111
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description: scientific article; zbMATH DE number 934260