Exploration of Soliton Solutions and Multi-Solitonic forms for the Extended Estevez-Prada and the Generalized P-Type Equations in (3+1)-Dimensions using the Advanced Mathematical Method

Abstract

This article extracts the newly developed soliton solutions of two higher-dimensional nonlinear partial differential equations (NLPDEs): the extended Estevez-Prada and the generalized P-type equations in (3+1)-dimensions. Both governing equations are Painlevé integrable and describe various complex dynamical phenomena in tsunami wave propagation, nonlinear dynamics, optical bullet dynamics in nonlinear fibers, and ion-acoustic soliton stability in magnetized plasmas. Using the extended Jacobian elliptic function expansion (EJEFE) method, we present novel families of closed-form solutions, with solitons, kinks, lumps, and their interaction structures in both equations. These solutions degenerate into trigonometric and hyperbolic functions once the modulus parameter approaches 0 and 1, respectively. Furthermore, we also analyze the dynamic structure of the solutions obtained, including solitons, lumps, kinks, and their interactions via 3D, contour, and 2D graphics. The results demonstrate the capability of the applied methods in obtaining solutions to higher-order NLPDEs and emphasize the need to explore the realms of computational mathematics and engineering further. This paper presents an original perspective on evolving multi-soliton, multi-lumps, and multi-peakon patterns by connecting plasma physics, theoretical physics, and nonlinear physical applications. These findings pave the path for future improvements in ocean waves and wave propagation, improving our understanding of complex systems.