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Symplectic nonsqueezing of the KdV flow

J. Colliander·M. Keel·G. Staffilani·H. Takaoka·T. Tao
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Abstract

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. Unlike the work of Kuksin 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. 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.

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J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. 2004-12-19. Symplectic nonsqueezing of the KdV flow. https://arxiv.org/abs/math/0412381

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