3028 articles  [version française]
 HAL: hal-00398799, version 1
 arXiv: 0906.4540
 The Szegö Cubic Equation
 (2009-06-24)
 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.
 1: Laboratoire de Mathématiques d'Orsay (LM-Orsay) CNRS : UMR8628 – Université Paris XI - Paris Sud 2: Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO) Université d'Orléans – CNRS : UMR7349
 Subject : Mathematics/Complex VariablesMathematics/Analysis of PDEs
 Keyword(s): Nonlinear Schrödinger equations – Integrable Hamiltonian systems – Lax pairs – Hankel operators
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 hal-00398799, version 1 http://hal.archives-ouvertes.fr/hal-00398799 oai:hal.archives-ouvertes.fr:hal-00398799 From: Sandrine Grellier <> Submitted on: Wednesday, 24 June 2009 19:33:13 Updated on: Wednesday, 24 June 2009 20:33:33