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Zbl 1101.35070
Grahovski, Georgi G.; Gerdjikov, Vladimir S.; Kostov, Nikolay A.; Atanasov, Victor A.
New integrable multi-component NLS type equations on symmetric spaces: $\Bbb Z_4$ and $\Bbb Z_6$ reductions. (English)
[A] Mladenov, Iva\"ilo (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. 154-175 (2006). ISBN 954-8495-30-9/pbk

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.

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
37K30 Relations with algebraic structures
82B10 Quantum equilibrium statistical mechanics (general)

Keywords: Lie algebras; Bose-Einstein condensates; Lax operators

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	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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