On the soliton sensitivity, lump and analytic solutions of the generalized (3+1)-dimensional P-type nonlinear wave equation

In this manuscript, we investigated a generalized (3+1)-dimensional P-type nonlinear wave equation. First, for a more detailed analysis of the studied model, we transformed it into a nonlinear ordinary differential equation by utilizing an appropriate wave transformation formula. We utilized a modified Riccati equation method (also called the generalized Riccati equation method), signifying its first application for this model. The extraction of traveling waves and singular solitons used exponential and hyperbolic trigonometric functions, which exhibited features of darkness, brightness, and mixed states. The bell-shaped soliton, or solitary soliton solution, and the rational lump solution were found. To enhance the understanding of our findings, we incorporated numerical simulations in both two- and three-dimensional representations, effectively illustrating the physical implications of our results. The modified Riccati sub-equation function method solutions extended prior results by providing explicit closed-form expressions and a systematic analysis of the role of the dispersion parameter $ \alpha_4 $, which had not been previously quantified for this model. Additionally, we confirmed that all solutions are valid; they were substituted back into their respective partial differential equation and satisfied the required conditions.