Traveling waves for the cubic Szego equation on the real line

TL;DR

This paper characterizes all traveling wave solutions of the cubic Szego equation on the real line as rational functions with one pole and demonstrates their orbital stability, contrasting with the instability observed on the circle.

Contribution

It provides a complete classification of traveling waves for the cubic Szego equation on the real line and proves their orbital stability, a novel result compared to previous circle case studies.

Findings

Traveling waves are rational functions with a single pole.

All such traveling waves are orbitally stable.

Contrast with instability results on the circle S^1.

Abstract

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.