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New integrable multi-component NLS type equations on symmetric spaces: ${ℤ}_{4}$ and ${ℤ}_{6}$ 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 $\mathrm{𝔰𝔭}\left(4\right)$, $\mathrm{𝔰𝔭}\left(10\right)$ and $\mathrm{𝔰𝔬}\left(12\right)$ Lie algebras. The MNLS related to $\mathrm{𝔰𝔭}\left(4\right)$ is a three-component MNLS which finds applications to Bose-Einstein condensates. The MNLS related to $\mathrm{𝔰𝔬}\left(12\right)$ and $\mathrm{𝔰𝔬}\left(10\right)$ Lie algebras after convenient ${ℤ}_{6}$ or ${ℤ}_{4}$ reductions reduce to three and four-component MNLS showing new types of ${\chi }^{\left(3\right)}$-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:
 35Q55 NLS-like (nonlinear Schrödinger) equations 37K30 Relations of infinite-dimensional systems with algebraic structures 82B10 Quantum equilibrium statistical mechanics (general)
##### Keywords:
Lie algebras; Bose-Einstein condensates; Lax operators