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The Szegö Cubic Equation

Abstract : 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.
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Contributor : Sandrine Grellier <>
Submitted on : Wednesday, June 24, 2009 - 7:33:13 PM
Last modification on : Monday, December 23, 2019 - 3:50:10 PM
Document(s) archivé(s) le : Tuesday, June 15, 2010 - 6:40:48 PM


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  • HAL Id : hal-00398799, version 1
  • ARXIV : 0906.4540



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



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