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arXiv: 0906.4540

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The Szegö Cubic Equation

Patrick Gérard 1, Sandrine Grellier 2

(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 Variables Mathematics/Analysis of PDEs

Keyword(s): Nonlinear Schrödinger equations – Integrable Hamiltonian systems – Lax pairs – Hankel operators

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From: Sandrine Grellier <>

Submitted on: Wednesday, 24 June 2009 19:33:13

Updated on: Wednesday, 24 June 2009 20:33:33