Invariant tori for the cubic Szegö equation

Abstract : We continue the study of the following Hamiltonian equation on the Hardy space of the circle, $i\partial _tu=\Pi(|u|^2u)\ ,$ where $\Pi$ denotes the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. In a previous work, we proved that this equation admits a Lax pair, and that it is completely integrable. In this paper, we construct the action-angle variables, which reduces the explicit resolution of the equation to a diagonalisation problem. As a consequence, we solve an inverse spectral problem for Hankel operators. Moreover, we establish the stability of the corresponding invariant tori. Furthermore, from the explicit formulae, we deduce the classification of orbitally stable and unstable traveling waves.
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https://hal.archives-ouvertes.fr/hal-00538601
Contributor : Sandrine Grellier <>
Submitted on : Tuesday, November 23, 2010 - 10:54:02 AM
Last modification on : Monday, December 23, 2019 - 3:50:10 PM
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• HAL Id : hal-00538601, version 1
• ARXIV : 1011.5479

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Patrick Gérard, Sandrine Grellier. Invariant tori for the cubic Szegö equation. Inventiones Mathematicae, Springer Verlag, 2012, 187, pp.707-754. ⟨hal-00538601⟩

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