The First Integral of the Dissipative Nonlinear Schrödinger Equation with Nucci’s Direct Method and Explicit Wave Profile Formation

The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the (1+1)− dimensional Schrödinger dynamical equation, which subject this study. New solitary wave are discovered using Nucci’s scheme a new extended direct algebraic method. approach provides an easy general mechanism for covering 37 solitonic solutions, roughly corresponds to all families, reduction method used develop first integral exact solution partial differential equations. Thus, there several patterns that obtained, including plane solution, mixed hyperbolic periodic trigonometric shock singular complex solutions. considered model obtained utilizing scheme. We present 2-D, 3-D, contour graphics results illustrate pulse characteristics while taking suitable values parameters involved, we observed influence on waves. It noticed number α speed μ responsible controlling amplitude periodicity propagating solution.