Theoretical and Mathematical Physics

, Volume 192, Issue 2, pp 1097–1114 | Cite as

Kulish–Sklyanin-type models: Integrability and reductions

  • V. S. GerdjikovEmail author


We start with a Riemann–Hilbert problem (RHP) related to BD.I-type symmetric spaces SO(2r + 1)/S(O(2r − 2s+1) ⊗ O(2s)), s ≥ 1. We consider two RHPs: the first is formulated on the real axis R in the complexplane; the second, on RiR. The first RHP for s = 1 allows solving the Kulish–Sklyanin (KS) model; the second RHP is related to a new type of KS model. We consider an important example of nontrivial deep reductions of the KS model and show its effect on the scattering matrix. In particular, we obtain new two-component nonlinear Schrödinger equations. Finally, using the Wronski relations, we show that the inverse scattering method for KS models can be understood as generalized Fourier transforms. We thus find a way to characterize all the fundamental properties of KS models including the hierarchy of equations and the hierarchy of their Hamiltonian structures.


symmetric space multicomponent nonlinear Schrödinger equation Lax representation reduction group 


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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Institute for Advanced Physical StudiesNew Bulgarian UniversitySofiaBulgaria
  3. 3.Institute for Nuclear Research and Nuclear EnergyBulgarian Academy of SciencesSofiaBulgaria

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