Vol. 4, No. 3, 2011

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 itu = Π(|u|2u) in the Hardy space L+2() 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 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.

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