The cubic Szegő equation and Hankel operators

By Sandrine Grellier, Patrick Gérard. Astérisque, 2015

89 citations recorded by the source corpus.

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 Szego 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 equation with 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 $s>\frac 12$ 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.

View Paper: https://www.semanticscholar.org/paper/d291c28bdaba075980fe6b0d7ec226a178f7a2df