Explicit formula for the solution of the Szegö equation on the real line and applications

Abstract

We consider the cubic Szego equationi 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 &gt;= 0, for generic rational function data. As for nongeneric data, we construct an example for which soliton resolution holds only in H(s), 0 +/-infinity parallel to u(t)parallel to H(s) = infinity ; s &gt; 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).</p