Analytical solutions and modulation instability analysis to the perturbed nonlinear Schrödinger equation

The perturbed nonlinear Schrödinger equation, describing an ultra-short optical pulse propagate along the parabolic law nonlinear fibers with Raman effect and self-steepening, has been investigated using the sub-equation expansion method. The dark and singular optical solitons are obtained. Furthermore, the modulational instability (MI) is analyzed based on the standard linear-stability analysis. The MI gain is got. These results have important application features in the area of telecommunications.