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