# La structure symplectique de l'espace de phase de l'équation Korteweg-de Vries périodique

Bättig, D; Bloch, A; Guillot, J C; Kappeler, T (1993). La structure symplectique de l'espace de phase de l'équation Korteweg-de Vries périodique. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 317(11):1019-1022.

## Abstract

e prove that the generalized phase space of the Korteweg-de Vries equation on S1, i.e., (L20([0,1]),ωG), where ωG is the Gardner symplectic structure on the space L20([0,1]) of L2 functions with mean zero, is symplectomorphic to the phase space (l21/2(R2),ω0) of infinitely many harmonic oscillators, where l21/2(R2) is the Hilbert space of sequences (xn,yn)n≥1 satisfying ∑n≥1n(x2n+y2n)<∞ endowed with the canonical symplectic structure ω0. The symplectomorphism Ω from (L20([0,1]),ωG) onto (l21/2(R2),ω0) is shown to be bianalytic. Similar results hold for the periodic Toda equations and the periodic nonlinear Schrödinger equation.

## Abstract

e prove that the generalized phase space of the Korteweg-de Vries equation on S1, i.e., (L20([0,1]),ωG), where ωG is the Gardner symplectic structure on the space L20([0,1]) of L2 functions with mean zero, is symplectomorphic to the phase space (l21/2(R2),ω0) of infinitely many harmonic oscillators, where l21/2(R2) is the Hilbert space of sequences (xn,yn)n≥1 satisfying ∑n≥1n(x2n+y2n)<∞ endowed with the canonical symplectic structure ω0. The symplectomorphism Ω from (L20([0,1]),ωG) onto (l21/2(R2),ω0) is shown to be bianalytic. Similar results hold for the periodic Toda equations and the periodic nonlinear Schrödinger equation.