Abstract:

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. In this talk we discuss the explicit formula for the solutions of the Szegö equation. We also present the construction of generalized action-angle coordinates for the Szegö equation restricted to finite dimensional manifolds of rational functions. As an application, we solve the inverse spectral problem for Hankel operators of finite rank (whose symbols are rational functions). If times allows, we present two other applications: the soliton resolution of certain solutions and an example of solution whose high Sobolev norms grow to infinity over time.