Analysis & PDE

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

Traveling waves for the cubic Szegő equation on the real line

Oana Pocovnicu

Full-text: Open access

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(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

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

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

Permanent link to this document
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

Subjects
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]

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

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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References

  • N. Burq, P. Gérard, and N. Tzvetkov, “Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces”, Invent. Math. 159:1 (2005), 187–223.
  • T. Cazenave and P.-L. Lions, “Orbital stability of standing waves for some nonlinear Schrödinger equations”, Comm. Math. Phys. 85:4 (1982), 549–561.
  • W. Eckhaus and P. Schuur, “The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions”, Math. Methods Appl. Sci. 5:1 (1983), 97–116.
  • P. Gérard, “Description du défaut de compacité de l'injection de Sobolev”, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233.
  • P. Gérard and S. Grellier, “The cubic Szegő equation”, Ann. Sci. Éc. Norm. Supér. $(4)$ 43:5 (2010), 761–810.
  • P. Gérard and S. Grellier, “L'équation de Szegö cubique”, in Séminaire X Équations aux dérivées partielles (Palaiseau, 2008), École Polytechnique, 2010.
  • T. Hmidi and S. Keraani, “Remarks on the blowup for the $L\sp 2$-critical nonlinear Schrödinger equations”, SIAM J. Math. Anal. 38:4 (2006), 1035–1047.
  • J. Holmer and M. Zworski, “Soliton interaction with slowly varying potentials”, Int. Math. Res. Not. 2008:10 (2008), Art. ID rnn026.
  • L. H örmander, The analysis of linear partial differential operators, I: Distribution theory and Fourier analysis, 2nd ed., Grundlehren der Math. Wiss. 256, Springer, Berlin, 1990.
  • P. D. Lax, “Translation invariant spaces”, Acta Math. 101 (1959), 163–178.
  • N. K. Nikolski, Operators, functions, and systems: an easy reading, I: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs 92, American Mathematical Society, Providence, RI, 2002.
  • V. V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer, New York, 2003.
  • G. Perelman, “A remark on soliton-potential interactions for nonlinear Schrödinger equations”, Math. Res. Lett. 16:3 (2009), 477–486.
  • M. Reed and B. Simon, Methods of modern mathematical physics, III: Scattering theory, Academic Press, New York, 1979.
  • W. Rudin, Real and complex analysis, 2nd ed., McGraw-Hill, New York, 1974.
  • M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates”, Comm. Math. Phys. 87:4 (1982), 567–576.