## 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 language | English (US) |
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Pages (from-to) | 376-397 |

Number of pages | 22 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 97 |

Issue number | 4 |

DOIs | |

State | Published - 1996 |

Externally published | Yes |

## 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