Traveling waves for the cubic Szegő equation on the real line

Pocovnicu, Oana

doi:10.2140/apde.2011.4.379

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Traveling waves for the cubic Szegő equation on the real line

Oana Pocovnicu

Vol. 4 (2011), No. 3, 379–404

DOI: 10.2140/apde.2011.4.379

Abstract

We consider the cubic Szegő equation $i \partial t u = Π (| u |^{2} u)$ 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 $p \in ℂ$ with $Im p < 0$ . 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

Vol. 4, No. 3, 2011