Abstract
We investigate the generalized nonlinear Schrödinger equation (NLSE) of third order analytically, that accept the single-parameter family of one hump embedded soliton. This equation has been utilized to model ultra-short pulses in optical fibers. The physical phenomenon of this dynamical equation is revealed through the exact solutions for example solitons, elliptic function solutions, etc. These types of novel explicit solutions are constructed of generalized third-order NLSE via the modified extended direct algebraic method (MEDAM), that have key applications in engineering and applied sciences. Movements of some achieved solutions are described graphically, that assists to know the physical interpretation of this equation. Furthermore, the configuration conditions of dark-bright solitons are also presented. The influence of this scheme is shown in computational work and achieved results.