# The symplectic structure of the phase space for the periodic Korteweg-de Vries equation

Bättig, D; Bloch, A; Guillot, J-C; Kappeler, T (1993). The symplectic structure of the phase space for the periodic Korteweg-de Vries equation. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics, 317(11):1019-1022.

## Abstract

We prove that the generalized phase space of KdV on S 1 , i.e. (L 0 2 ([0,1]),ω G ) where ω G denotes the Gardner symplectic structure on the space L 0 2 ([0,1]), of L 2 functions with average 0, is symplectomorphic to the phase space (l 1/2 2 (ℝ 2 ),ω 0 ) of infinitely many harmonic oscillators, where l 1/2 2 (ℝ 2 ) denotes the Hilbert space of sequences (x n ,y n ) n≥1 satisfying ∑ n≥1 n(x n 2 +y n 2 )<∞ endowed with the canonical symplectic structure ω 0 . The symplectomorphism Ω from (L 0 2 ([0,1],ω G ) onto (l 1/2 2 (ℝ 2 ),ω 0 ) is shown to be bianalytic. Similar results hold for the periodic Toda equations and the periodic nonlinear Schrödinger equation (defocusing).

## Abstract

We prove that the generalized phase space of KdV on S 1 , i.e. (L 0 2 ([0,1]),ω G ) where ω G denotes the Gardner symplectic structure on the space L 0 2 ([0,1]), of L 2 functions with average 0, is symplectomorphic to the phase space (l 1/2 2 (ℝ 2 ),ω 0 ) of infinitely many harmonic oscillators, where l 1/2 2 (ℝ 2 ) denotes the Hilbert space of sequences (x n ,y n ) n≥1 satisfying ∑ n≥1 n(x n 2 +y n 2 )<∞ endowed with the canonical symplectic structure ω 0 . The symplectomorphism Ω from (L 0 2 ([0,1],ω G ) onto (l 1/2 2 (ℝ 2 ),ω 0 ) is shown to be bianalytic. Similar results hold for the periodic Toda equations and the periodic nonlinear Schrödinger equation (defocusing).

## Additional indexing

Other titles: La structure symplectique de l‘espace de phase de l‘équation Korteweg-de-Vries périodique Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Gardner symplectic structure; symplectomorphism; periodic Toda equations; periodic nonlinear Schrödinger equation French 1993 29 Nov 2010 16:29 05 Apr 2016 13:28 Elsevier 0764-4442 http://www.ams.org/mathscinet-getitem?mr=1249781

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