Symplectic nonsqueezing of the Korteweg-de Vries flow
type: Item
instance of: scholarly article
title: Symplectic nonsqueezing of the Korteweg-de Vries flow
zbMATH Open document ID: 1105.35104
DOI: 10.1007/BF02588080
author: Markus Keel
author: Gigliola Staffilani
author: Hideo Takaoka
author: James Colliander
published in: Acta Mathematica
publication date: 2006-12-12T00:00:00Z
review text: Let \(u(t)\) denote the solution of the KdV equation \[ u_{t}+u_{xxx}=6uu_{x}, \quad u(0,x)=u_{0}(x), \] on the circle \(x\in{\mathbf{T}}\). The nonlinear flow map \(S_{KdV}(t)\) on \({C}^{\infty}(\mathbf{T})\) is defined by \(S_{KdV}(t) u_{0}:=u(t).\) It is uniformly continuous on \(H_{0}^{s}\) for \(s \geq{} -\tfrac{1}{2}\). In order to apply results from symplectic geometry, the infinite-dimensional flow must first be approximated by a finite-dimensional flow. In the first step, \(S_{KdV}\) is approximated by \(S_{BKdV}\) in the strong \(H_{0}^{s}\) topology. This is proved by using the Miura transform. Let \(\omega\) be a symplectic form on the Hilbert space \(\mathbf{H}\). We say that \((\mathbf{H},\omega)\) is the symplectic phase space of a PDE with Hamiltonean \(H[u(t)]\) if the PDE can be written in the form \[ \frac{du}{dt}=J\nabla{H[u(t)]}. \] Here \(J\) is an almost complex structure on \(\mathbf{H}\) which is compatible with the Hilbert space inner product. The final result of this paper is the following statement for the infinite-dimensional ball \(\mathbf{B}^{\infty}\) in the phase space \(H_{0}^{-\tfrac{1}{2}}(\mathbf{T})\). Theorem. Let \(0<r<R,\;u_{\star}\in{H_{0}^{-\tfrac{1}{2}}(\mathbf{T})},\;k_{0}\in{\mathbb{Z}^{\star}},\;z\in\mathbb{C}\) and \(T>0\). Then \[ S_{KdV}(T)(\mathbf{B}^{\infty}(u_{\star};R))\not\subseteq \mathbf{C}_{k_{0}}^{\infty}(z;r). \]
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K25
Mathematics Subject Classification ID: 47H20
zbMATH DE Number: 5079885
zbMATH Keywords: KdV flow
zbMATH Keywords: symplectic behaviour
zbMATH Keywords: Sobolev space
zbMATH Keywords: Hamiltonean
zbMATH Keywords: Miura transform
MaRDI profile type: Publication
cites work: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II: The KdV-equation
cites work: Q4697831
cites work: Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations
cites work: Periodic Korteweg-de Vries equation with measures as initial data
cites work: Q4255122
cites work: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations
cites work: Global well-posedness for KdV in Sobolev spaces of negative index
cites work: A Refined Global Well-Posedness Result for Schrödinger Equations with Derivative
cites work: Sharp global well-posedness for KdV and modified KdV on ℝ and 𝕋
cites work: Multilinear estimates for periodic KdV equations, and applications
cites work: Q4003380
cites work: Korteweg-de Vries Equation and Generalizations. IV. The Korteweg-de Vries Equation as a Hamiltonian System
cites work: Pseudo holomorphic curves in symplectic manifolds
cites work: Q4304742
cites work: Q5483289
cites work: Q5693352
cites work: The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices
cites work: A bilinear estimate with applications to the KdV equation
cites work: On the ill-posedness of some canonical dispersive equations.
cites work: Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDE's
cites work: A simple model of the integrable Hamiltonian equation
cites work: Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation
cites work: Q4273954
cites work: Q4650706
description: scientific article; zbMATH DE number 5079885