Long-time asymptotics of the nonlinear Schrödinger equation shock problem


Journal Article

The long-time asymptotics of two colliding plane waves governed by the focusing nonlinear Schrödinger equation are analyzed via the inverse scattering method. We find three asymptotic regions in space-time: a region with the original wave modified by a phase perturbation, a residual region with a one-phase wave, and an intermediate transition region with a modulated two-phase wave. The leading-order terms for the three regions are computed with error estimates using the steepest-descent method for Riemann-Hilbert problems. The nondecaying initial data requires a new adaptation of this method. A new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions. Also, the effect of the collision is felt in the plane-wave state well beyond the shock front at large times. © 2007 Wiley Periodicals, Inc.

Full Text

Duke Authors

Cited Authors

  • Buckingham, R; Venakides, S

Published Date

  • September 1, 2007

Published In

Volume / Issue

  • 60 / 9

Start / End Page

  • 1349 - 1414

Electronic International Standard Serial Number (EISSN)

  • 0010-3640

International Standard Serial Number (ISSN)

  • 0010-3640

Digital Object Identifier (DOI)

  • 10.1002/cpa.20179

Citation Source

  • Scopus