Vol. 4, No. 3, 2011

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Traveling waves for the cubic Szegő equation on the real line

Oana Pocovnicu

Vol. 4 (2011), No. 3, 379–404
Abstract

We consider the cubic Szegő equation $i\partial tu=\Pi \left(|u{|}^{2}u\right)$ in the Hardy space ${L}_{+}^{2}\left(ℝ\right)$ on the upper half-plane, where $\Pi$ is the Szegő projector. It was first introduced by Gérard and Grellier as a toy model for totally nondispersive evolution equations. We show that the only traveling waves are of the form $C∕\left(x-p\right)$, where $p\in ℂ$ with $Im\phantom{\rule{0.3em}{0ex}}p<0$. Moreover, they are shown to be orbitally stable, in contrast to the situation on the unit disk where some traveling waves were shown to be unstable.

Keywords
nonlinear Schrödinger equations, Szegő equation, integrable Hamiltonian systems, Lax pair, traveling wave, orbital stability, Hankel operators
Mathematical Subject Classification 2000
Primary: 35B15, 37K10, 47B35