Astérisque, n° 389. The cubic Szego equation and Hankel operators

This monograph is devoted to the dynamics on Sobolev spaces of the cubic Szeg(...) equation on the circle (...), (...) Here II denotes the orthogonal projector from (...) onto the subspace (...) of functions with nonnegative Fourier modes. We construct a nonlinear Fourier transformation on (...) allowing to describe explicitly the solutions of this equation with data in (...). This explicit description implies almost-periodicity of every solution in this space. Furthermore, it allows to display the following turbulence phenomenon. For a dense (...) subset of initial data in (...), the solutions tend to infinity in Hs for every (...) 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.