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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.
MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K30Relations of infinite-dimensional systems with algebraic structures
53C35Symmetric spaces (differential geometry)
35Q51Soliton-like equations