Symplectic non-squeezing for the KdV flow on the line

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.