Symplectic nonsqueezing of the Korteweg-de Vries flow
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 \[ S_{KdV}(T)(\mathbf{B}^{\infty}(u_{\star};R))\not\subseteq \mathbf{C}_{k_{0}}^{\infty}(z;r). \]
scientific article; zbMATH DE number 5079885