Our work aims to investigate the vcBLMPE in ( 3 + 1 ) -dimensions (3D-vcBLMPE) that characterizes wave propagation in incompressible fluids. In real-world issues, nonlinear partial differential equations containing time-dependent coefficients are more relevant than those with constant coefficients owing to inhomogeneities of media and nonuniformities of boundaries. In shallow water, linearization of the wave formation needs more critical wave capacity criteria than in water depths, and the strongly nonlinear aspects are readily visible. By using symbolic computation, several nonautonomous wave solutions with different geometric structures are obtained. Each of the gained solutions is presented graphically based on various arbitrary coefficients to demonstrate and better comprehend their dynamical properties. As a comparison between the new results and the results previously reported, we have presented several completely new findings in this study.