New bright and dark soliton solutions for a generalized nonlinear Schrödinger equation

In this paper, we obtain new dark and bright soliton solutions of the generalized nonlinear Schrödinger equation that describes the ultrashort pulses propagate through a nonlinear medium. Using a traveling wave ansatz, the nonlinear Schrödinger equation is transformed into two nonlinear ordinary differential equations. These two ordinary differential equations are proved to be equivalent under certain conditions. And the solutions of these equations are obtained through a direct reduction to a first order ordinary differential equation with well-known solutions. The obtained solutions are in the form of Jacobi elliptic functions that degenerate to dark and bright soliton solutions. The new soliton solutions obtained in this paper can assist us with understanding the propagation of solitons through a nonlinear medium.