Traveling and soliton waves and their characteristics in the extended (3+1)-dimensional Kadomtsev–Petviashvili equation in fluid

In this work, the extended (3+1)-dimensional Kadomtsev–Petviashvili model, which explains the development of nonlinear, long waves with tiny amplitudes and a gradual dependency on the transverse coordinate, is solved in terms of traveling waves. In order to conduct the investigation, the modified extended direct algebraic approach is used. Numerous unique methods for traveling waves are provided. Solitons of the dark, bright, and singular waves are among these solutions. In addition, Weierstrass elliptic, Jacobi elliptic, and singular periodic wave solutions are provided as well. To demonstrate the potency and nature of the raised solutions, the 3D graphical representation and the density graphs are introduced.