Nonlinear wave dynamics of fractional unstable and modified unstable nonlinear Schrödinger equations through analytical solutions

In this article, we investigate the behavior of unstable NLSE and the modify unstable NLSE to derive solitonic solutions. These equations describe the time evolution of disturbances in an unstable medium. We employ the amended extended tangent hyperbolic function method to solve these equations, yielding precise solutions that are simple, rational, and hyperbolic. We also establish restricted conditions to ensure the existence of these solutions. The behaviors elucidated encompass hyperbolic, periodic, kink, dark, and combo dark bright waves, as well as other complex phenomena. To enhance understanding and visual appeal, we generate contour, density, 2D, and 3D plots of select solutions by appropriately choosing parameters. Our proposed solutions aim to elucidate the intricacies of the model under consideration, catering to researchers interested in exploring its nuances. The outcomes of our study reaffirm the efficacy and significance of the employed method. Moreover, we demonstrate its robustness and applicability to a broad spectrum of other unstable nonlinear mathematical physics problems.