On spectral theory of Lax operators on symmetric spaces: vanishing versus constant boundary conditions. (English) Zbl 1205.35292

Gerdjikov, Vladimir S.

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].

Reviewer: Q. P. Liu (Beijing)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q15	Riemann-Hilbert problems in context of PDEs
78A46	Inverse problems (including inverse scattering) in optics and electromagnetic theory
35A08	Fundamental solutions to PDEs
81U40	Inverse scattering problems in quantum theory

Keywords:

NLS; symmetric space; inverse sacttering transformation; Riemann-Hilbert problem

Citations:

Zbl 1180.37105