We show symplectic nonsqueezing for the KdV equation on the line
.
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 Vişan (Ann. of Math.190:1 (2019), 249–305) in the well-posedness theory of the equation in
low-regularity spaces, relying on its completely integrable structure.
The employment of our methods also provides us with a new concise
proof of symplectic nonsqueezing for the same equation on the circle
,
recovering the result of Colliander et al. (Acta Math. 195 (2005), 197–252).
