On the multi-component NLS type equations on symmetric spaces: reductions and soliton solutions. (English) Zbl 1079.53075

Mladenov, Ivaïlo M.(ed.) et al., Proceedings of the 6th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 3–10, 2004. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-9-5/pbk). 203-217 (2005).
This is an introductory survey on multi-component nonlinear Schrödinger equations related to simple Lie algebras and symmetric spaces. These generalizations of the nonlinear Schrödinger equation were proposed by Fordy and Kulish in the early 1990’s. Recently, the authors sytematically studied these equations, their reductions and soliton solutions [Inverse Probl. 17, No. 4, 999–1015 (2001; Zbl 0988.35143) and J. Phys. A, Math. Gen. 34, No. 44, 9425–9461 (2001; Zbl 1001.37074)]. In particular, some of their results on this subject are reviewed and discussed in this paper.
For the entire collection see [Zbl 1066.53003].


35Q55 NLS equations (nonlinear Schrödinger equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
53C35 Differential geometry of symmetric spaces
35Q51 Soliton equations