Riemann–Hilbert problem for the focusing nonlinear Schrödinger equation with multiple high-order poles under nonzero boundary conditions

The Riemann–Hilbert (RH) problem is developed to study the focusing nonlinear Schrödinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace the residues at the simple- and the second-poles. Furthermore, the solution of RH problem is transformed into a closed system of algebraic equations, and the soliton solutions corresponding to the transmission coefficient 1/s11(z) with an N-order pole are obtained by solving the algebraic system. Then, in a more general case, the transmission coefficient with multiple high-order poles is studied, and the corresponding solutions are obtained. In addition, for high-order pole, the propagation behavior of the soliton solution corresponding to a third-order pole and the mixed case of a second-order pole and a simple pole are given as example. •The transmission coefficient 1/s11(z) with arbitrary order poles is studied.•A general expression of the solution corresponding to a higher-order pole is given.•The expression of the solution corresponding to multiple higher-order poles is also presented.•The dynamics of soliton solutions corresponding to mixed higher-order poles are analyzed.•The results are general and can be applied to other AKNS models.