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New integrable multi-component NLS type equations on symmetric spaces: $\Bbb Z_4$ and $\Bbb Z_6$ reductions. (English) Zbl 1101.35070
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 7th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 2--10, 2005. Sofia: Bulgarian Academy of Sciences (ISBN 954-8495-30-9/pbk). 154-175 (2006).
Summary: The reductions of the multi-component nonlinear Schrödinger models related to C.I and D.III type symmetric spaces are studied. We pay special attention to the MNLS related to the ${\germ{sp}}(4)$, ${\germ{sp}}(10)$ and ${\germ{so}}(12)$ Lie algebras. The MNLS related to ${\germ{sp}}(4)$ is a three-component MNLS which finds applications to Bose-Einstein condensates. The MNLS related to ${\germ{so}}(12)$ and ${\germ{so}}(10)$ Lie algebras after convenient $\bbfZ_6$ or $\bbfZ_4$ reductions reduce to three and four-component MNLS showing new types of $\chi^{(3)}$-interactions that are integrable. We briefly explain how these new types of MNLS can be integrated by the inverse scattering method. The spectral properties of the Lax operators $L$ and the corresponding recursion operator $\Lambda$ are outlined. Applications to spinor model of Bose-Einstein condensates are discussed. For the entire collection see [Zbl 1089.53004].

35Q55NLS-like (nonlinear Schrödinger) equations
37K30Relations of infinite-dimensional systems with algebraic structures
82B10Quantum equilibrium statistical mechanics (general)