## Proceedings of the International Conference on Geometry, Integrability and Quantization

### 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.

#### Article information

Dates
First available in Project Euclid: 13 July 2015

https://projecteuclid.org/ euclid.pgiq/1436792597

Digital Object Identifier
doi:10.7546/giq-7-2006-154-175

Mathematical Reviews number (MathSciNet)
MR2228370

Zentralblatt MATH identifier
1101.35070

#### Citation

Grahovski, Georgi G.; Gerdjikov, Vladimir S.; Kostov, Nikolay A.; Atanasov, Victor A. New Integrable Multi-Component NLS Type Equations on Symmetric Spaces: $\mathbb{Z}_4$ and $\mathbb{Z}_6$ Reductions. Proceedings of the Seventh International Conference on Geometry, Integrability and Quantization, 154--175, Softex, Sofia, Bulgaria, 2006. doi:10.7546/giq-7-2006-154-175. https://projecteuclid.org/euclid.pgiq/1436792597