Invariant tori for the cubic Szeg\"o equation

We continue the study of the following Hamiltonian equation on the Hardy space of the circle, $$i\partial _tu=\Pi(|u|^2u)\ ,$$ where $\Pi$ denotes the Szeg\"o projector. This equation can be seen as a toy model for totally non dispersive evolution equations. In a previous work, we proved that this equation admits a Lax pair, and that it is completely integrable. In this paper, we construct the action-angle variables, which reduces the explicit resolution of the equation to a diagonalisation problem. As a consequence, we solve an inverse spectral problem for Hankel operators. Moreover, we establish the stability of the corresponding invariant tori. Furthermore, from the explicit formulae, we deduce the classification of orbitally stable and unstable traveling waves.