Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem

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Abstract

In this paper we consider the semi-classical limit of the non-self-adjoint Zakharov-Shabat eigenvalue problem. We conduct a series of careful numerical experiments which provide strong evidence that the number of eigenvalues scales like ε-1, just as in the self-adjoint case, and that the eigenvalues appear to approach a limiting curve. One general choice of potential functions produces a Y-shaped spectrum. We give an asymptotic argument which predicts a critical value for the phase for which the straight line spectra bifurcates to produce the Y-shaped spectra. This asymptotic prediction agrees quite well with numerical experiments. The asymptotic argument also predicts a symmetry breaking in the eigenfunctions, which we are able to observe numerically. We also show that the number of eigenvalues living away from the real axis for a restricted class of potentials is bounded by cε-1, where c is an explicit constant. A complete theory of the shape of the eigenvalue curve and a general bound on the number of eigenvalues is still lacking.

Original languageEnglish (US)
Pages (from-to)376-397
Number of pages22
JournalPhysica D: Nonlinear Phenomena
Volume97
Issue number4
DOIs
StatePublished - 1996
Externally publishedYes

Keywords

  • Inverse scattering method
  • Semiclassical limits
  • Zakharov-Shabat eigenvalue problem

ASJC Scopus subject areas

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

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