Exploring the chaotic behavior, and ion acoustic wave of generalized perturbed Korteweg-de Vries equation with a fractional operator

This manuscript studies the chaotic behaviors and ion acoustic wave of the M-fractional generalized perturbed Korteweg-de Vries equation. Here we explain some assertions of the M-fractional operator. The proposed model is a mathematical framework that characterizes the evolution of waves within a nonlinear and dispersive medium. Chaotic behaviors are analyzed using bifurcation theory, where different perturbation terms are introduced. To illustrate the dynamics, we visualize the phase portrait through 3-D and 2-D plots, time series, and Poincaré diagrams. Additionally, we utilize the modified Sardar sub-equation technique to investigate the solitary ion acoustic wave solutions for the M-fractional generalized perturbed Korteweg-de Vries (M-fGPKdV) equation. Under some conditions, the solutions are expressed as the trigonometric, exponential, and hyperbolic functions form. We'll demonstrate these solutions’ unique structure and fascinating interaction behaviour. We'll also use graphs (3-D, 2-D and density plots) to discuss the dynamics of the results after setting the parameters to the proper values. These findings contribute to a deeper understanding of the dynamics of the M-fGPKdV equation and its applications in real-world phenomena.