Traveling waves for the cubic Szegő equation on the real line

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Abstract

We consider the cubic Szego equation i partial derivative tu = Pi(vertical bar u vertical bar(2) u) in the Hardy space L-+(2) (R) on the upper half-plane, where Pi is the Szego projector. It was first introduced by Gerard 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 epsilon 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.

Original languageEnglish
Pages (from-to)379-404
Number of pages26
JournalAnalysis and PDE
Volume4
Issue number3
DOIs
Publication statusPublished - 2011

Keywords

  • nonlinear Schrodinger equations
  • Szego equation
  • integrable Hamiltonian systems
  • Lax pair
  • traveling wave
  • orbital stability
  • Hankel operators
  • NONLINEAR SCHRODINGER-EQUATIONS

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