New integrable multi-component NLS type equations on symmetric spaces: \(\mathbb Z_4\) and \(\mathbb Z_6\) reductions. (English) Zbl 1101.35070

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

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 \({\mathfrak{sp}}(4)\), \({\mathfrak{sp}}(10)\) 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 \(\mathbb{Z}_6\) or \(\mathbb{Z}_4\) 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.
For the entire collection see [Zbl 1089.53004].

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
37K30	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
82B10	Quantum equilibrium statistical mechanics (general)

Keywords:

Lie algebras; Bose-Einstein condensates; Lax operators