Kulish–Sklyanin-type models: Integrability and reductions

We start with a Riemann–Hilbert problem (RHP) related to BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s+1)\otimes O(2s))$, $s\ge1$. We consider two RHPs: the first is formulated on the real axis $\mathbb R$ in the complex-$\lambda$ plane; the second, on $\mathbb R\oplus i\mathbb R$. 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.