New Integrable Multi-Component NLS Type Equations on Symmetric Spaces: Z_4 and Z_6 Reductions

The reductions of the multi-component nonlinear Schrodinger (MNLS) type models related to C.I and D.III type symmetric spaces are studied. We pay special attention to the MNLS related to the sp(4), so(10) and so(12) Lie algebras. The MNLS related to sp(4) is a three-component MNLS which finds applications to Bose-Einstein condensates. The MNLS related to so(12) and so(10) Lie algebras after convenient Z_6 or Z_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.