Notes on symplectic non-squeezing of the KdV flow

We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle T. The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results are similar to those obtained for the NLS flow by Bourgain [3]. A major difficulty here though is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by a finite-dimensional Hamiltonian flow. To resolve this problem we invert the Miura transform and work on the level of the modified KdV (mKdV) equation, for which smoothing estimates can be established. J.C. was supported in part by N.S.E.R.C. Grant RGPIN 250233-03 and the Sloan Foundation. M.K. was supported in part by N.S.F. Grant DMS 9801558. G.S. was supported in part by N.S.F. Grant DMS 0100345 and by a grant from the Sloan Foundation. H.T. was supported in part by J.S.P.S. Grant No. 13740087. T.T. was a Clay Prize Fellow and was supported in part by grants from the Packard Foundation.