Advanced Mathematical Approaches for the Kadomtsev–Petviashvili and Bogoyavlensky–Konopelchenko Equations in Applied Sciences

The Kadomtsev–Petviashvili (KP) equation and the Bogoyavlensky–Konopelchenko (BK) equation are fundamental models in the study of nonlinear wave dynamics, describing the evolution of weakly dispersive, quasi‐two‐dimensional (2D) wave phenomena in integrable systems. In this article, we introduce a novel analytical technique, the ‐expansion method, designed to derive exact, closed‐form solutions to these equations with increased efficiency and generality. The KP equation, which describes the propagation of surface waves in shallow water or plasma waves in a cylindrical geometry, and the BK equation, a higher‐dimensional generalization of the KP equation, are both critical in understanding soliton dynamics and wave interactions in nonlinear media. By exploiting the structure of the equations and the interplay between various terms, the method enables the construction of exact solutions that are difficult to obtain using traditional perturbation or ansatz‐based approaches. We apply this method to derive several classes of solutions to both the KP and BK equations, including multisoliton solutions, complex wave structures, and exact traveling wave solutions. Our results highlight the flexibility of the method in capturing a wide range of solution types, which are highly relevant to real‐world applications, such as wave pattern formation, soliton interactions, and stability analysis in nonlinear systems. Using the proposed expansion method, innovative solutions are derived, including an antibell‐shaped soliton, a kink‐shaped soliton, and a singular periodic solution. These results are presented through three‐dimensional (3D), 2D, and contour plots, offering a clear understanding of their physical properties.