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

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

2010-12-14

arxiv.org/abs/1012.2943v2

Mathematics

Analysis of PDEs

Small modifications in the proof of Proposition 1.3, changed the order in the proof of Theorem 1.9, replaced the proof of \chi

Scientific paper

We consider the cubic Szeg\"o equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szeg\"o projector on positive frequencies. 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 Szeg\"o equation. As an application, we prove soliton resolution in H^s for all s>0, for generic data. As for non-generic data, we construct an example for which soliton resolution holds only in H^s, 01/2. 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 Szeg\"o equation with generic data are spirals around Lagrangian toroidal cylinders T^N \times R^N.

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