Invariant tori for the cubic Szeg\"o equation

TL;DR

This paper studies the cubic Szeg"o equation on the Hardy space, constructing action-angle variables, solving an inverse spectral problem, and classifying the stability of traveling waves, thereby advancing understanding of its integrable structure.

Contribution

It constructs explicit action-angle variables for the cubic Szeg"o equation, solves the inverse spectral problem for Hankel operators, and classifies the stability of traveling waves.

Findings

Explicit action-angle variables are constructed.

Inverse spectral problem for Hankel operators is solved.

Classification of stable and unstable traveling waves is provided.

Abstract

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.