© 2022 World Scientific Publishing Company.In this paper, the novel exact solitary wave solutions for the generalized nonlinear Schrödinger equation with parabolic nonlinear (NL) law employing the improved cosh(Γ(ϖ)) -sech(Γ(ϖ)) function scheme and the combined cos(Γ(ϖ)) -sec(Γ(ϖ)) function scheme are found. Diverse collections of hyperbolic and trigonometric function solutions acquired rely on a map between the considered equation and an auxiliary ODE. Received solutions are recast in several hyperbolic, rational and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solution. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model utilizing mathematical software. The main subject of this work is that one can visualize and update the knowledge to overcome the most common techniques and defeat to solve the ODEs and PDEs. We demonstrated that these solutions validated the program using Maple and found them correct. The proposed methodology for solving the metamaterials model has been designed to be effectual, unpretentious, expedient and manageable. Applications of the solutions by the mentioned techniques will be useful to investigate the signals properties of optical fibers, plasma physics phenomena, electromagnetic fields occurrences and various types of nonlinear metamaterials models.