Fundamental Analytic Solutions for the Kulish-Sklyanin Model with Constant Boundary Conditions

Abstract

In the present paper we analyze the construction of fundamental analytic solutions (FAS) for the generalized Kulish-Sklyanin models (KSM) for vanishing (VBC) and constant boundary conditions (CBC). Using FAS one can reduce the direct and inverse scattering problems for the Lax operator to a Riemann-Hilbert problem (RHP). For VBC we find two FAS $\chi^+(x,t,\lambda) $ and $\chi^-(x,t,\lambda) $ analytic in the upper/lower $\mathbb{C}_\pm$ complex $\lambda$-plane. The RHP consists in: given the sewing function $G(x,t,\lambda)$ to constructing both $\chi^\pm(x,t,\lambda) $ in their regions of analyticity. For CBC the problem becomes more complicated, because now the RHP must be formulated on a Riemannian surface of genus 1.