EXPLICIT FORMULA FOR THE SOLUTION OF THE SZEGO EQUATION ON THE REAL LINE AND APPLICATIONS

We consider the cubic Szego equation i partial derivative(t)u = Pi (vertical bar u vertical bar(2)u) in the Hardy space L-+(2) ( R) on the upper half-plane, where Pi is the Szego projector. 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 Szego equation. As an application, we prove soliton resolution in H-s for all s >= 0, for generic rational function data. As for nongeneric 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 in finity over time, i. e. lim(t) > +/-infinity parallel to u(t)parallel to H-s = infinity ; 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 Szego equation with generic rational function data are spirals around Lagrangian toroidal cylinders T-N x R-N.