Multiform Solitary Wave Structures and Bifurcation Analysis of an Extended Modified (3+1)-Dimensional Kadomtsev–Petviashvili Equation via Hybrid Analytical Approaches
Abstract
This work presents the derivation of exact analytical solutions for a (3+1)-dimensional modified Kadomtsev–Petviashvili equation using two effective analytical schemes, namely the Improved Simple Equation Method and the exponential expansion approach. The considered model is relevant for describing complex nonlinear wave propagation in incompressible fluid environments, where dispersive and nonlinear effects are strongly coupled. The implemented techniques yield a rich variety of wave structures with distinct physical characteristics, including singular and non-singular localized formations, periodic patterns, and exponentially decaying profiles. In addition, hyperbolic and trigonometric configurations are obtained, reflecting different propagation regimes of the system. The qualitative behavior of these solutions is further clarified through three-dimensional surface representations and contour mappings. To ensure the physical admissibility of the obtained solutions, a detailed stability investigation is carried out, highlighting their robustness under perturbations. The findings confirm that the adopted analytical frameworks are highly efficient in constructing diverse nonlinear wave forms and provide deeper insight into the governing mechanisms of multidimensional wave evolution.
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