## Analysis & PDE

• Anal. PDE
• Volume 4, Number 3 (2011), 379-404.

### Traveling waves for the cubic Szegő equation on the real line

Oana Pocovnicu

#### Abstract

We consider the cubic Szegő equation $i∂tu=Π(|u|2u)$ in the Hardy space $L+2(ℝ)$ on the upper half-plane, where $Π$ 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 $C∕(x−p)$, where $p∈ℂ$ with $Imp<0$. 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.

#### Article information

Source
Anal. PDE, Volume 4, Number 3 (2011), 379-404.

Dates
Revised: 28 April 2010
Accepted: 29 May 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731146

Digital Object Identifier
doi:10.2140/apde.2011.4.379

Mathematical Reviews number (MathSciNet)
MR2872121

Zentralblatt MATH identifier
1270.35172

#### Citation

Pocovnicu, Oana. Traveling waves for the cubic Szegő equation on the real line. Anal. PDE 4 (2011), no. 3, 379--404. doi:10.2140/apde.2011.4.379. https://projecteuclid.org/euclid.apde/1513731146

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