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

TL;DR

This paper develops the construction of fundamental analytic solutions for the generalized Kulish-Sklyanin models under both vanishing and constant boundary conditions, enabling the reduction of scattering problems to Riemann-Hilbert problems.

Contribution

It introduces a method to construct FAS for KSM with constant boundary conditions, extending the RHP formulation to a genus 1 Riemann surface.

Findings

FAS are constructed for VBC with analytic properties in complex plane

RHP formulation is extended to CBC on a Riemann surface of genus 1

The approach simplifies solving the inverse scattering problem for KSM

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.