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Invariant tori for the cubic Szegö equation

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Abstract

We continue the study of the following Hamiltonian equation on the Hardy space of the circle,

$$i\partial_tu=\Pi(|u|^2u),$$

where Π 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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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628, Université Paris-Sud XI, 91405, Orsay, France

    Patrick Gérard

  2. Fédération Denis Poisson, MAPMO-UMR 6628, Département de Mathématiques, Université d’Orleans, 45067, Orléans Cedex 2, France

    Sandrine Grellier

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  1. Patrick Gérard
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  2. Sandrine Grellier
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Correspondence to Sandrine Grellier.

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P. Gérard is member of the Institut Universitaire de France.

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Gérard, P., Grellier, S. Invariant tori for the cubic Szegö equation. Invent. math. 187, 707–754 (2012). https://doi.org/10.1007/s00222-011-0342-7

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