Strong NLS soliton-defect interactions

Roy H. Goodman, Philip J. Holmes, Michael I. Weinstein

Research output: Contribution to journalArticlepeer-review

148 Scopus citations

Abstract

We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et al. [Phys. Rev. E 49 (1994) 3400], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finite-dimensional model, compare them with the nonlinear Schrödinger simulations, and comment on differences due to the finite-dimensional ansatz.

Original languageEnglish (US)
Pages (from-to)215-248
Number of pages34
JournalPhysica D: Nonlinear Phenomena
Volume192
Issue number3-4
DOIs
StatePublished - Jun 1 2004

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Keywords

  • Collective coordinates
  • Hamiltonian systems
  • Nonlinear scattering
  • Periodic orbits
  • Resonant energy transfer
  • Stable manifolds
  • Two-mode model

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