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

Georgi G. Grahovski, Vladimir S. Gerdjikov, Nikolay A. Kostov, and Victor A. Atanasov

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

Source
Proceedings of the Seventh International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov and Manuel de León, eds. (Sofia: Softex, 2006), 154-175

Dates
First available in Project Euclid: 13 July 2015

Permanent link to this document
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


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