Explicit formula for the solution of the Szeg

By Oana Pocovnicu. 2010

57 citations recorded by the source corpus.

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, 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 Szeg\"o equation with generic data are spirals around Lagrangian toroidal cylinders T^N \times R^N.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/73d7c061cbb46e5ea0fcfc7938dc7b89d7c13a68

Citations in the corpus

• An explicit formula for the cubic Szegö equation (2013, unknown): “Indeed, in [12], Pocovnicu obtains an explicit formula for all rational solutions of the cubic Szeg˝o equation on the line, and displays a non generic rational solution of this equation such that, for every s > This illustrates how the growing phenomenon of high Sobolev norms may differ depending on…”