The Szegö Cubic Equation

Abstract

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i{\partial }_{t}u=\Pi (|u{|}^{2}u),$$ 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.