Notes on symplectic non-squeezing of the KdV flow

Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.

Journées équations aux dérivées partielles (2005), article no. 14, 15 p.

Nous montrons deux résultats d’approximation de dimension finie et une propriété « nonsqueezing » symplectique pour le flot Korteweg-de Vries (KdV) sur le cercle $𝕋$. Le résultat nonsqueezing dépend des résultats d’approximation mentionnés et du théorème nonsqueezing de Gromov en dimension finie. Contrairement aux travaux de Kuksin [22] qui a lancé l’étude de résultats nonsqueezing pour des systèmes hamiltoniens de dimension infinie, l’argument nonsqueezing ici ne construit pas de capacité de façon directe. De cette manière, nos résultats sont semblables à ceux obtenus pour le flot NLS par Bourgain [3]. Cependant, une difficulté majeure ici est le manque d’estimations de lissage qui nous permettraient d’approximer facilement le flot KdV de dimension infinie par un flot hamiltonien de dimension finie. Pour contourner ce problème, nous inversons la transformation de Miura et travaillons au niveau de l’équation KdV modifiée (mKdV), pour laquelle une estimation de lissage peut être obtenue.

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.

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