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

Abstract

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