In this current article, we have discussed the solitary wave (3+1)-dimensional Mikhailov–Novikov–Wang Integrable (MNWI) equation analytically and graphically. The governing MNWI equation has been used to derive the various types of analytical soliton and periodic wave solutions, distinguished by their nonlinear wave propagation and invariant shape, including velocity. In addition, a heuristic solution for the (3+1)-dimensional MNWI equation is obtained through the Hirota bilinear method (HBM) in terms of the analysis of one soliton, two solitons, three solitons, breather first order, and the interaction between first-order breather and one-soliton wave solutions. It has been found that the nonlinear wave propagation maintains its invariant shape and velocity. The results demonstrate how solitary wave interactions aid in the spread of solitary wave phenomena. The propagation of solitary wave phenomena refers to the stable, self-reinforcing travel of localized waves through a medium, maintaining their shape and energy balance due to the interplay between nonlinearity and dispersion. The outcomes of the current work will be helpful in modeling floods, tsunamis, and flow in significant reversals, among other environmental and physical conditions.