Construction of new wave patterns for the (3+1)-dimensional Kadomtsev–Petviashvili equation using a couple of integrating architectures

The aim of this study is to extract new exact solutions for the Kadomtsev–Petviashvili (KP) equation in three spatial dimensions and one time dimension, which is widely used in quantum field theory and plasma physics. This research refers to two separate methods–the generalized Arnous (GA) method and the ${\displaystyle \frac{G^{\mathrm{\prime }}}{b{G}^{\mathrm{\prime }}+G+a}}$ expansion method to uncover various new soliton solutions for the given model. These solutions are usually expressed as rational, trigonometric, and hyperbolic functions, which increase the usefulness of the method for practical applications. In addition, the conditions certifying that these solutions remain valid are also identified. The behavior of these solutions is shown through a visual representation. The recorded results are new and show how both methods are effective and robust, making them valuable tools for solving various differential equations in applied sciences and engineering.