The Szegö Cubic Equation

Patrick Gérard (1) , Sandrine Grellier (2)

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

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i\partial _tu=\Pi(|u|^2u)\ ,$$ where $\Pi $ is the Szegö 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.

Mots clés: Nonlinear Schrödinger equations; Integrable Hamiltonian systems; Lax pairs; Hankel operators

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

Patrick Gérard, Sandrine Grellier. The Szegö Cubic Equation. Annales Scientifiques de l'École Normale Supérieure, 2010, 43, pp.761-809. ⟨hal-00398799⟩