The cubic Szegő equation

Gérard, Patrick; Grellier, Sandrine

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 5, pp. 761-810.

On considère l’équation hamiltonienne suivante sur l’espace de Hardy du cercle

$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$

où $\Pi$ désigne le projecteur de Szegő. Cette équation est un cas modèle d’équation sans aucune propriété dispersive. On établit qu’elle admet une paire de Lax et une infinité de lois de conservation en involution, et qu’elle peut être approchée par une suite de systèmes hamiltoniens de dimension finie complètement intégrables. Néanmoins, on met en évidence des phénomènes d’instabilité illustrant la dégénérescence de cette structure complètement intégrable. Enfin, on caractérise les ondes progressives de ce système.

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle,

$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$

where $\Pi$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system.

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