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arXiv · 0906.4540

The Szegö Cubic Equation

Patrick Gérard·Sandrine Grellier
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Abstract

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i\partial_tu=Π(|u|^2u) ,$$ where $Π$ 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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Patrick Gérard, Sandrine Grellier. 2009-06-24. The Szegö Cubic Equation. https://arxiv.org/abs/0906.4540

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