Kulish-Sklyanin type models: integrability and reductions

Abstract

We start with a Riemann-Hilbert problem (RHP) related to a BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s +1)\otimes O(2s))$, $s\geq 1$. We consider two Riemann-Hilbert problems: the first formulated on the real axis $\mathbb{R}$ in the complex $\lambda$-plane; the second one is formulated on $\mathbb{R} \oplus i\mathbb{R}$. The first RHP for $s=1$ allows one to solve the Kulish-Sklyanin (KS) model; the second RHP is relevant for a new type of KS model. An important example for nontrivial deep reductions of KS model is given. Its effect on the scattering matrix is formulated. In particular we obtain new 2-component NLS equations. Finally, using the Wronskian relations we demonstrate that the inverse scattering method for KS models may be understood as a generalized Fourier transforms. Thus we have a tool to derive all their fundamental properties, including the hierarchy of equations and the hierarchy of their Hamiltonian structures.