The cubic szego equation and hankel operators

TL;DR

This paper provides an explicit nonlinear Fourier transform for the cubic Szeg{"o} equation on the circle, revealing solution behaviors including almost-periodicity and turbulence phenomena with unbounded growth in Sobolev norms.

Contribution

It introduces a novel nonlinear Fourier transformation based on spectral data of Hankel operators, enabling explicit solution descriptions and analysis of complex dynamics.

Findings

Solutions are almost-periodic in H^{1/2}_+.

There exists turbulence with solutions exhibiting super-polynomial growth in Sobolev norms.

Certain solutions return close to initial data after large times.

Abstract

This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1.It is devoted to the dynamics on Sobolev spaces of the cubic Szeg{\"o} equation on the circle ${\mathbb S} ^1$,$$ i\partial \_t u=\Pi (\vert u\vert ^2u)\ .$$Here $\Pi $ denotes the orthogonal projector from $L^2({\mathbb S} ^1)$ onto the subspace $L^2\_+({\mathbb S} ^1)$ of functions with nonnegative Fourier modes.We construct a nonlinear Fourier transformation on $H^{1/2}({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$ allowing to describe explicitly the solutions of this equationwith data in $H^{1/2}({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$. This explicit description implies almost-periodicity of every solution in $H^{\frac 12}\_+$. Furthermore, it allows to display the following turbulence phenomenon. For a dense $G\_\delta $ subset of initial data in $C^\infty ({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$, the solutions tend to infinity in $H^s$ for every $s\textgreater{}\frac 12$ with super--polynomial growth on some sequence of times, while they go back to their initial data on another sequence of times tending to infinity. This transformation is defined by solving a general inverse spectral problem involving singular values of a Hilbert--Schmidt Hankel operator and of its shifted Hankel operator.