An analytical exploration of dark and bright solitary wave solutions of time-fractional (3+1) -dimensional generalized Painlevé-type equation

This research study constructs and examines solitary wave solutions to the Katugampola time-fractional Formula: see text-dimensional generalized Painlev'e-type (also known as P-type) equation with the help of the generalized Formula: see text-expansion method and tanh-coth method. The targeted model is extensively used to understand the dynamics of the plasma waves and instabilities in soliton theory, plasma physics and nonlinear wave theory. By offering closed form traveling solutions, the proposed methods transform the desired model into a nonlinear algebraic system. A new set of solutions in terms of trigonometric, rational and hyperbolic functions is obtained by analytically solving the resulting algebraic system using the symbolic computation tool Maple. We provide a range of 3D, density, and 2D visual representations that show the presence of bright and dark solitary wave solutions in the framework of the resulting wave profiles that illustrate the strong dynamics of the model. Also, a more dynamic and practical framework proposed by Katugampola-derivative is used to enhance the validity of the results by investigating how the time-fractional derivative affects the given model's solutions. Being the initial accounts of developing solutions of the aimed model, our comparison brings out the uniqueness of the obtained solutions. The variegated array of findings also shows that the suggested approaches are useful mathematical tools for solving nonlinear Fractional Partial Differential Equations (FPDEs), with applications in mathematical sciences such as engineering, physics, and biology, as well as in other nonlinear evolution problems.