The cubic Szegő equation and Hankel operators

This monograph is devoted to the dynamics on Sobolev spaces of the cubic Szegő equation on the circle ^1, equation* i_t u=(u^2u). equation* Here denotes the orthogonal projector from L^2(^1) onto the subspace L^2_+(^1) of functions with nonnegative Fourier modes. We construct a nonlinear Fourier transformation on H^1/2(^1)L^2_+(^1) allowing to describe explicitly the solutions of this equation with data in H^1/2(^1)L^2_+(^1). This explicit description implies almost-periodicity of every solution in this space. Furthermore, it allows to display the following turbulence phenomenon. For a dense G_ subset of initial data in C^(^1)L^2_+(^1), the solutions tend to infinity in H^s for every s>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.