New Integrable Multi-Component NLS Type Equations on Symmetric Spaces: $\mathbb{Z}_4$ and $\mathbb{Z}_6$ Reductions

Abstract

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 $\mathfrak{sp}(4)$, $\mathfrak{so}(10)$ and $\mathfrak{so}(12)$ Lie algebras. The MNLS related to $\mathfrak{sp}(4)$ is a three-component MNLS which finds applications to Bose–Einstein condensates. The MNLS related to $\mathfrak{so}(12)$ and $\mathfrak{so}(10)$ Lie algebras after convenient $\mathfrak{Z}_6$ or $\mathfrak{Z}_4$ 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.