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.

17 pages, 2 Figures

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• Dark solitons in F=1 spinor Bose--Einstein condensate
• Inverse scattering method for square matrix nonlinear Schrödinger equation under nonvanishing boundary conditions
• New Integrable Multi-Component NLS Type Equations on Symmetric Spaces: Z_4 and Z_6 Reductions