Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili $\left(\mathrm{KP}\right)$ model within a (3+1)-dimensional domain, governing the behavior of wave propagation in a medium characterized by both nonlinearity and dispersion. The inquiry employs two distinct analytical techniques to derive multiple soliton solutions and multiple solitary wave solutions. These methods include the modified Sardar sub-equation technique and the Darboux transformation $\left(\mathrm{DT}\right)$. The modified Sardar sub-equation technique is used to obtain multiple soliton solutions, while the $\mathrm{DT}$ is introduced to develop two bright and two dark soliton solutions. These solutions are presented alongside their corresponding constraint conditions and illustrated through 3-D, 2-D, and contour plots to physically portray the derived solutions. The results demonstrate that the employed analytical techniques are useful and have not yet been explored in the context of the analyzed models. The proposed methodologies are valuable and can be applied to additional nonlinear evolutionary models employed to describe nonlinear physical models within the domain of nonlinear science.