The cubic Szegő equation

Patrick Gérard; Sandrine Grellier

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 5, page 761-810
  • ISSN: 0012-9593

Abstract

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We consider the following Hamiltonian equation on the L 2 Hardy space on the circle, i ∂ t u = Π ( | u | 2 u ) , where Π is the Szegő 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.

How to cite

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Gérard, Patrick, and Grellier, Sandrine. "The cubic Szegő equation." Annales scientifiques de l'École Normale Supérieure 43.5 (2010): 761-810. <http://eudml.org/doc/272217>.

@article{Gérard2010,
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 Szegő 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.},
author = {Gérard, Patrick, Grellier, Sandrine},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {nonlinear schrödinger equations; integrable hamiltonian systems; Lax pairs; Hankel operators},
language = {eng},
number = {5},
pages = {761-810},
publisher = {Société mathématique de France},
title = {The cubic Szegő equation},
url = {http://eudml.org/doc/272217},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Gérard, Patrick
AU - Grellier, Sandrine
TI - The cubic Szegő equation
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 5
SP - 761
EP - 810
AB - 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 Szegő 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.
LA - eng
KW - nonlinear schrödinger equations; integrable hamiltonian systems; Lax pairs; Hankel operators
UR - http://eudml.org/doc/272217
ER -

References

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