Comprehensive study of a (3+1)-dimensional nonlinear Vakhnenko-Parkes dynamical equation with applications in nonlinear wave propagation in relaxing media

This research conducted a study of the nonlinear (3 + 1)-dimensional Vakhnenko-Parkes (VP) equation since it acts as a vital model for high-frequency wave perturbations in relaxing high-rate active barotropic media. The analytical solutions emerged through the modified auxiliary equation method and the improved F-expansion method which serve as advanced tools for studying nonlinear waves. These methods present diverse solutions that contain solitary waves combined with periodic waves and rational forms. The obtained solutions exhibit their behavior through illustrated 2D and 3D plots showing how waves evolve and how their structures transform with varying parameters. This visual analysis shows how solutions disperse and stay stable thus making them relevant for fluid dynamics investigations and wave propagation studies. The analytical understanding of nonlinear wave models in high-frequency barotropic systems receives new insights through our results which enhance mathematical and physical VP equation descriptions.