Nonsqueezing property of the coupled KdV type system without Miura transform

Abstract

We prove the nonsqueezing property of the coupled Korteweg-de Vries (KdV) equation. Relying on Gromov's nonsqueezing theorem for finite dimensional Hamiltonian systems, the argument is to approximate the solutions to the original infinite dimensional Hamiltonian system by a frequency truncated finite dimensional system, and then the nonsqueezing property is transferred to the infinite dimensional system. This is the argument used by Bourgain for the 1D cubic NLS flow, and Colliander et. al. for the KdV flow. One of main ingredients of \cite{Colliander:2005vv} is to use the Miura transform to change the KdV flow to mKdV flow. In this work, we consider the coupled KdV equations for which the Miura transform is not available. Instead of the Miura transform, we use the method of the normal form via the differentiation by parts. Although we present the proof for the coupled KdV equation, the same proof is applicable to the KdV flow, and so provide alternative simplified proof for the KdV flow.

Mathematical error pointed by an anonymous reviewer is corrected. We improved exposition

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