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The Riemann-Hilbert (RH) problem is first developed to study the focusingnonlinear Schr\"{o}dinger (NLS) equation with multiple high-order poles undernonzero boundary conditions. Laurent expansion and Taylor series are employedto replace the residues at the simple- and the second-poles. Further, thesolution of RH problem is transformed into a closed system of algebraicequations, and the soliton solutions corresponding to the transmissioncoefficient $1/s_{11}(z)$ with an $N$-order pole are obtained by solving thealgebraic system. Then, in a more general case, the transmission coefficientwith multiple high-order poles is studied, and the corresponding solutions areobtained. In addition, for high-order pole, the propagation behavior of thesoliton solution corresponding to a third-order pole is given as example.

Riemann-Hilbert problem for the nonlinear Schr\"{o}dinger equation with multiple high-order poles under nonzero boundary conditions