The Szeg\"o Cubic Equation

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i{\mathrm{\partial }}_{t}u=\mathrm{\Pi }(\mathrm{\mid }u{\mathrm{\mid }}^{2}u),$$ where $\mathrm{\Pi }$ is the Szeg"o 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.