# American Institute of Mathematical Sciences

September  2011, 31(3): 607-649. doi: 10.3934/dcds.2011.31.607

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

 1 Laboratoire de Mathématiques d’Orsay, Université Paris-Sud (XI), 91405, Orsay, France

Received  January 2010 Revised  April 2011 Published  August 2011

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$.
Citation: Oana Pocovnicu. Explicit formula for the solution of the Szegö equation on the real line and applications. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 607-649. doi: 10.3934/dcds.2011.31.607
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##### References:
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