On the symplectic structure of the phase space for periodic KdV, Toda, and defocusing NLS
type: Item
instance of: scholarly article
title: On the symplectic structure of the phase space for periodic KdV, Toda, and defocusing NLS
zbMATH Open document ID: 0855.58035
DOI: 10.1215/S0012-7094-95-07914-9
published in: Duke Mathematical Journal
publication date: 1997-02-11T00:00:00Z
review text: Let \(L^2_0= \{q\in C^2[ 0,1 ]\); \(\int^1_0 q(x) dx= 0\}\) be a space of periodic potentials of average 0. According to Gardner, the phase space \((L^2_0, \omega)\) for the KdV equation can be identified with the space of potentials \((p, q)\) from \(L^2_0\) with symplectic structure \[ \omega (p, q)= \sum_{n\geq 1} (\pi n)^{-1} (p_{2n}\wedge q_{2n-1}- p_{2n-1} \wedge q_{2n}), \] where \(p_n\) and \(q_n\) are Fourier modes for \(p\) and \(q\): \[ q(x)= 2\sum_{n\geq 1} q_{2n} \cos 2\pi nx+ q_{2n-1} \sin 2\pi nx. \] The main result of the paper is an implicit construction of symplectomorphisms \(\Omega: (L^2_0, \omega)\to (\ell^2_{1/2} (\mathbb{R}^2), \omega_0)\) where \(\ell^2_{1/2} (\mathbb{R}^2)\) consists of the sequences \((x_n, y_n)\), \(n\geq 1\), \(\sum n (x^2_n+ y^2_n)< \infty\), \(\omega_0= \sum_n (x_n \wedge y'_n- y_n \wedge x'_n)\). Moreover, in polar coordinates \(r_j, \alpha_j\) given by \(x_j= r_j \cos \alpha_j\), \(y_j= r_j \sin \alpha_j\), the quantities \({1\over2} r^2_j, \alpha_j\) are action-angle variables for the KdV equation. This theorem, besides the proof of the existence of global action-angle variables, allows to describe the structure of singularities for the fibration of the phase space by invariant tori. Analogous considerations are given for defocusing nonlinear Schrödinger equations and periodic Toda lattices.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 878408
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: symplectomorphism
zbMATH Keywords: action-angle variables
zbMATH Keywords: nonlinear Schrödinger equations
zbMATH Keywords: periodic Toda lattices
author: Daniel Bättig
author: Anthony M. Bloch
author: Jean-Claude Guillot
author: T. Kappeler
reviewed by: Sergei Khoroshkin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 878408