In this article, the Fordy-Gibbons-Jimbo-Miwa equation is analyzed, a special form of the Kadomtsev-Petviashvili hierarchy equation, which is one of the most prominent nonlinear dynamical models with two spatial and a temporal coordinate that represents the evolution of long, nonlinear, small-amplitude waves with a gradual dependence on the transverse coordinate. The governing model is investigated analytically by employing the extended generalized Riccati equation mapping approach (GREM). Furthermore, the dynamics of several wave structures are visualized in 3D, 2D, and contour forms for a given set of parameters using Mathematica 13.0 to demonstrate their characteristics, which has been achieved by selecting appropriate values of the relevant parameters. These solutions exhibit the characteristics of v-shaped, singular, and multi-bell-shaped, singular periodic, and multi-periodic solitons. Additionally, it has been confirmed that the model under consideration is a stable nonlinear structure by validating the established results. A range of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics and other relevant domains and research areas can be solved using these approaches, as demonstrated by their simplicity, clarity, and effectiveness, as well as the computational complexities and results.
