#### Vol. 4, No. 3, 2011

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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
##### Milestones
Received: 19 January 2010
Revised: 28 April 2010
Accepted: 29 May 2010
Published: 28 December 2011
##### Authors
 Oana Pocovnicu Laboratoire de Mathématiques d’Orsay Université Paris-Sud (XI) Campus d’Orsay, bât. 430 91405, Orsay France