Explicit formula for the solution of the Szeg\"o equation on the real line and applications

Abstract

We consider the cubic Szeg\"o equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szeg\"o projector on positive frequencies. It is a model for totally non-dispersive evolution equations and is completely integrable in the sense that it admits a Lax pair. We find an explicit formula for solutions of the Szeg\"o equation. As an application, we prove soliton resolution in H^s for all s>0, for generic data. As for non-generic data, we construct an example for which soliton resolution holds only in H^s, 0<s<1/2, while the high Sobolev norms grow to infinity over time, i.e. \lim_{t\to\pm\infty}|u(t)|_{H^s}=\infty if s>1/2. As a second application, we construct explicit generalized action-angle coordinates by solving the inverse problem for the Hankel operator H_u appearing in the Lax pair. In particular, we show that the trajectories of the Szeg\"o equation with generic data are spirals around Lagrangian toroidal cylinders T^N \times R^N.