About fifteen years ago, Patrick Gérard and I introduced the cubic Szegö equation$$\begin{aligned}i \partial_{t} u & =\Pi\left(|u|^{2} u\right), \quad u=u(x, t), \quad x \in \mathbb{T}, t \in \mathbb{R} \\u(x, 0) & =u_{0}(x) .\end{aligned}$$Here $\Pi$ denotes the Szegö projector which maps $L^{2}(\mathbb{T})$-functions into the Hardy space of $L^{2}(\mathbb{T})$-traces of holomorphic functions in the unit disc. It turned out that the dynamics of this equation were unexpected. This motivated us to try to understand whether the cubic Szegö equation is an isolated phenomenon or not. This talk is part of this project.
We consider a family of perturbations of the cubic Szegö equation and look for their traveling waves. Let us recall that traveling waves are particular solutions of the form$$u(x, t)=\mathrm{e}^{-i \omega t} u_{0}\left(\mathrm{e}^{-i c t} x\right), \quad \omega, c \in \mathbb{R}$$We will explain how the spectral analysis of some operators allows to characterize them.
From joint works with Patrick Gérard.[-]

The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.[-]

Patrick Gérard and I introduced the cubic Szegö equation around ten years ago as a toy model of a totally non dispersive degenerate Hamiltonian equation. Despite of the fact that it is a complete integrable system, we proved that this equation develops some cascades phenomena. Namely, for a dense set of smooth initial data, the Szegö solutions have unbounded high Sobolev trajectories, detecting transfer of energy from low to high frequencies. However, this dense set has empty interior and a lot of questions remain opened to understand turbulence phenomena. Among others, we would like to understand how interactions of Fourier coefficients interfere on it. In a recent work, Biasi and Evnin explore the phenomenon of turbulence on a one parameter family of equations which goes from the cubic Szegö equation to what they call the 'truncated Szegö equation'. In this latter, most of the Fourier mode couplings are eliminated. However, they prove the existence of unbounded trajectories for simple rational initial data. In this talk, I will explain how, paradoxically, the turbulence phenomena may be promoted by adding a damping term. Those results are closely related to an inverse spectral theorem we proved on the Hankel operators.[-]