A symbolic computation approach and its application to the Kadomtsev-Petviashvili equation in two (3+1)-dimensional extensions

This work examines the multi-rogue-wave solutions for the Kadomtsev-Petviashvili (KP) equation in form of two (3+1)-dimensional extensions, which are soliton equations, using a symbolic computation approach. This approach is stated in terms of the special polynomials developed through a Hirota bilinear equation. The first, second, and third-order rogue wave solutions are derived for these equations. The interaction of many rogue waves is illustrated by the multi-rogue waves. The physical explanations and properties of the obtained results are plotted for specific values of the parameters $\alpha$ and $\beta$ to understand the physics behind the huge (rogue) wave appearance. The figures are represented in three-dimensional, and the contour plots and the density are shown at different values of parameters. The obtained results are significant for showing the dynamic actions of higher-rogue waves in the deep ocean and nonlinear optical fibers.