This study provides a comprehensive analytical and graphical exploration of the solitary wave solutions to the (3 \(+\) 1)-dimensional Mikhailov–Novikov–Wang integrable (MNWI) equation. We thoroughly examine the solitary wave solutions, emphasising their nonlinear wave propagation, invariant shape and constant velocity. The MNWI equation is used to derive various analytical solutions, including soliton, periodic and rational wave solutions. Additionally, we obtain a heuristic solution for the ( \(3+1\) )-dimensional MNWI equation using the Hirota bilinear method, focussing on soliton wave dynamics. The analysis highlights both the mathematical framework and the physical implications of the solutions. By defining bounds on the system’s variables, we assess the overall stability through the Lyapunov function. Nonlinear wave propagation is shown to maintain stability, shape and velocity under bounded conditions. These findings confirm the essential properties and dynamics of solitons. Furthermore, the study reveals complex hybrid solutions through which wave interactions are studied. The outcomes of this work hold significant potential for modelling various physical and environmental phenomena, such as floods, tsunamis and large-scale fluid flows.
