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Under investigation in this work is an extended nonlinear Schr\"{o}dingerequation with nonzero boundary conditions, which can model the propagation ofwaves in dispersive media. Firstly, a matrix Riemann-Hilbert problem for theequation with nonzero boundary conditions at infinity is systematicallydiscussed. Then the inverse problems are solved through the investigation ofthe matrix Riemann-Hilbert problem. Therefore, the general solutions for thepotentials, and explicit expressions for the reflection-less potentials arepresented. Furthermore, we construct the simple-pole and double-pole solutionsfor the equation. Finally, the remarkable characteristics of these solutionsare graphically discussed. Our results should be useful to enrich and explainthe related nonlinear wave phenomena in nonlinear fields.

Inverse scattering transform of an extended nonlinear Schr\"{o}dinger equation with nonzero boundary conditions and its multisoliton solutions