Notes on symplectic non-squeezing of the KdV flow
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Bibliographic record
Abstract
We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝕋</mml:mi> </mml:math> . 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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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
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| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
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| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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