Analysis & PDE

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

Traveling waves for the cubic Szegő equation on the real line

Oana Pocovnicu

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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(xp), 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.

Article information

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

Received: 19 January 2010
Revised: 28 April 2010
Accepted: 29 May 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B15: Almost and pseudo-almost periodic solutions 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

nonlinear Schrödinger equations Szegő equation integrable Hamiltonian systems Lax pair traveling wave orbital stability Hankel operators


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.

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