On spectral theory of Lax operators on symmetric spaces: vanishing versus constant boundary conditions.

*(English)* Zbl 1205.35292
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 10th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6--11, 2008. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-323-531-5/pbk). 88-123 (2009).

The nonlinear Schrödinger equation is a well-known system in both mathematics and physics, its two-component extension, known as Manakov model, plays an important role in nonlinear optics. Further extensions include the multi-component NLS associated with symmetric spaces as developed by Fordy, Kulish and collaborators. The paper under review studies these NLS equations from the viewpoint of inverse scattering transformations. Keeping in mind both vanishing and constant boundary conditions, the author introduces a way to construct the fundamental analytic solutions and relates the scattering problem to a Riemann-Hilbert problem in order to study the resolvent of the Lax operator and the spectral properties. Several concrete examples are considered. Reprint of J. Geom. Symmetry Phys. 15, 1--41 (2009;

Zbl 1180.37105). For the entire collection see [

Zbl 1169.81004].

##### MSC:

35Q55 | NLS-like (nonlinear Schrödinger) equations |

37K15 | Integration of completely integrable systems by inverse spectral and scattering methods |

35Q15 | Riemann-Hilbert problems |

78A46 | Inverse scattering problems |

35A08 | Fundamental solutions of PDE |

81U40 | Inverse scattering problems (quantum theory) |