Oana Pocovnicu — Page 2

arXiv2011

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

We consider the cubic Szegö equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szegö 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 Szegö 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, 0 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 data are spirals around Lagrangian toroidal cylinders T^N \times R^N.

math.AP↗

arXiv2010

Traveling waves for the cubic Szego equation on the real line

We consider the cubic Szego equation i u_t=Pi(|u|^2u) on the real line, with solutions in the Hardy space on the upper half-plane, where Pi is the Szego projector onto the non-negative frequencies. This equation was recently introduced by P. Gerard and S. Grellier as a toy model for totally non-dispersive evolution equations. We show that the only traveling waves are rational functions with one simple pole. Moreover, they are shown to be orbitally stable, in contrast to the situation of the circle S^1 studied by the above authors, where some traveling waves were shown to be unstable.

math.AP↗