Wave Profile, Paul-Painlevé Approaches and Phase Plane Analysis to the Generalized (3+1)-Dimensional Shallow Water Wave Model

In this paper, the solitary wave solutions, the periodic type, and single soliton solutions are acquired. Here, the Hirota bilinear operator is employed to investigate single soliton, periodic wave solutions and the asymptotic case of periodic wave solutions. By utilizing symbolic computation and the applied method, generalized (3+1)-dimensional shallow water wave (GSWW) equation is investigated. The variational principle scheme to case periodic forms is studied. The (3+1)-GSWW model exhibits travelling waves, as shown by the research in the current paper. Through three-dimensional design, contour design, density design, and two-dimensional design using Maple, the physical features of single soliton and periodic wave solutions are explained all right. The findings demonstrate the investigated model’s broad variety of explicit solutions. As a result, exact solitary wave solutions to the studied issues, including solitary, single soliton, and periodic wave solution, are found. The phase plane is quickly examined after establishing the Hamiltonian function. The effects of wave velocity and other free factors on the wave profile are also investigated. It is shown that the approach is practical and flexible in mathematical physics. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering.