N-Triple-Pole Solitons in Matrix NLS Systems: Inverse Scattering Transform Under Nonzero Boundary Conditions

This work presents the first systematic development of the inverse scattering transform for matrix nonlinear Schrödinger equations in the case where the discrete spectrum has triple poles, under nonzero boundary conditions at infinity. These systems arise physically as reductions modeling spinor Bose-Einstein condensates with hyperfine spin F=1 and find applications in nonlinear optics. A uniformization variable is employed to map the underlying Riemann surface to the complex plane, enabling a complete characterization of the analyticity, symmetries, and asymptotic behaviors of the Jost functions and scattering data. Extending the established framework for simple and double poles, we show that rankP(x,t,zn)=3 requires a third-order zero of deta(z) at z=zn, while rankP(x,t,zn)=2 necessitates a fourth-order zero—a nontrivial feature absent in lower-order cases. The discrete spectrum for both rank configurations is fully characterized, and the full singular behavior near a triple pole is derived, respecting the quartet symmetry zn, zn*, vk02/zn, vk02/zn* imposed by the nonzero boundary conditions. Solving the resulting matrix Riemann-Hilbert problem with triple poles yields the potential reconstruction formula and, in the reflectionless case, explicit expressions for general N-triple-pole soliton solutions, with a detailed example for N=1 presented to illustrate the construction.