Symplectic non-squeezing for the KdV flow on the line

TL;DR

This paper proves symplectic non-squeezing for the KdV equation on the real line using finite-dimensional approximation, building on recent advances in low regularity well-posedness and integrable structure, and offers a new proof for the circle case.

Contribution

It introduces a novel finite-dimensional approximation approach to establish symplectic non-squeezing for KdV on the line, extending previous circle results.

Findings

Symplectic non-squeezing holds for KdV on $\,\mathbb R$.

A new concise proof for the circle case is provided.

Weak well-posedness in $H^{-1}(\mathbb R)$ is established.

Abstract

We show symplectic non-squeezing for the KdV equation on the line $\mathbb R$. This is achieved via finite-dimensional approximation. Our choice of finite-dimensional Hamiltonian system that effectively approximates the KdV flow is inspired by the recent breakthrough of Killip and Visan in the well-posedness theory of KdV in low regularity spaces, relying on its completely integrable structure. We also prove and exploit a weak well-posedness result for KdV in $H^{-1}(\mathbb R)$. Furthermore, the employment of our methods provides a new concise proof for the known result of symplectic non-squeezing for the same equation on the circle $\mathbb T$, first proved by Colliander, Keel, Staffilani, Takaoka, and Tao.