Multi-soliton solutions with arbitrary-order poles of the nonlinear Schrödinger equation in inhomogeneous optical fibers

Abstract The variable-coefficient nonlinear Schrödinger equation is investigated via the Riemann–Hilbert problem (RHP) with arbitrary-order poles under zero boundary conditions. The asymptotic behavior, analyticity, and symmetry of the Jost solutions and scattering coefficients are derived. In the reflectionless case, the formulas of the soliton solutions corresponding to single and multiple higher-order poles are derived via the Laurent expansion of the RHP. The method transforms the solution process into solving linear algebraic equations, avoiding complicated calculations and directly yielding the expressions for higher-order pole solutions corresponding to scattering data with N distinct poles. In addition, several special poles and their corresponding soliton solutions are presented explicitly, including one second-order pole, one third-order pole, two second-order poles, and other cases. The results verify that the associated Riemann–Hilbert approach is efficient for constructing higher-order localized solutions of integrable systems.