Invariant tori for the cubic Szegö equation

Patrick Gérard (1) , Sandrine Grellier (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

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.

Mots clés: systèmes complètement intégrables; coordonnées action-angle; opérateurs de Hankel

Domaines: Variables complexes [math.CV]; Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]

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