Article Contents
Article Contents

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

• We consider the cubic Szegö equation

$i\partial$$t$$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$.
Mathematics Subject Classification: 35B15, 37K10, 47B35.

 Citation:

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