Study of the modulational instability and miscellaneous soliton for metamaterials via three powerful schemes

In this paper, the exact solitary wave solutions for the generalized nonlinear (NL) Schrödinger equation with parabolic NL law employing the generalized (G'/G)expansion technique, the improved tan(φ/2)-expansion technique and the modified exp-function technique are acquired. Different sets of exponential function solutions are acquired relying on a map between the considered equation and an auxiliary ordinary differential equation (ODE). The obtained solutions are concluded in several of the hyperbolic and trigonometric forms based on diverse 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. For more analysis, the modulation instability (MI) analysis of the proposed model with normal derivatives is also carried out for parabolic NL law. The main aim of this research is that one can visualize and update the knowledge to overcome from the most common methods to solve the ODEs and partial differential equations (PDEs). We demonstrated that these solutions validated the program using Maple and found them to be correct. The proposed methodology for resolving NLPDEs has been designed to be effectual, unpretentious, expedient, and manageable. Finally, the existence of the solutions for the constraint conditions is also shown