Oana Pocovnicu: The Szegö equation and inverse spectral problems for Hankel operators

In this talk we consider the cubic Szegö equation: i u_t = Pi (|u|^2u) on the real line, where Pi is the Szegö projector on non-negative frequencies. This equation was recently introduced as a model of a non-dispersive non-linear PDE by P. Gerard (Paris-Sud University, Orsay, France) and S. Grellier (University of Orleans, France). Like 1-d cubic NLS and KdV, it is known to be completely integrable in the sense that it possesses a Lax pair structure. The operator L in the Lax pair is a Hankel operator. As a consequence, it turns out that a whole class of finite dimensional manifolds, consisting of rational functions, is invariant under the flow of the Szegö equation. In this talk we discuss the explicit formula for the solutions of the Szegö equation. If times allows, we present three applications of this formula: the soliton resolution of certain solutions, an example of solution whose high Sobolev norms grow to infinity over time, and the construction of explicit generalized action-angle coordinates on finite dimensional manifolds of rational functions. As a consequence, we solve the inverse spectral problem for Hankel operators whose symbol is a rational function.