Explicit Non-Traveling Wave Solutions for the (3 1)-Dimensional Nonlinear Vakhnenko–Parkes Equation Arising in Fluid Dynamics+

The present work aims to utilize two extended reduction strategies to the (3+1)–dimensional nonlinear Vakhnenko–Parkes equation and construct some novel exact non-traveling wave solutions. These approaches transform the main equation into several solvable reduced equations. In each strategy, we first isolate an unconstrained regime of the reduction in which the reduced equation holds identically for an arbitrary smooth reduced function; the exponential, trigonometric, hyperbolic, rational, Jacobi elliptic, and hybrid solution types are then obtained as particular choices of this general family. The solutions obtained from the proposed techniques are expressed in generalized forms that combine exponential, trigonometric, hyperbolic, rational, Jacobi elliptic, and hybrid wave structures. One of the main properties of the constructed solutions is the presence of arbitrary functions, which allows the solutions to possess greater flexibility in describing various phenomena in practical applications. Consequently, the resulting solutions generalize conventional traveling wave solutions and provide a flexible description of non-autonomous and spatially modulated nonlinear wave patterns. The reduction strategies employed generate several families of exact solutions with diverse functional forms. We subsequently include several graphical representations to illustrate the shape of the given solutions. The correctness of all solutions is verified by direct substitution into the governing equation. The proposed framework systematically constructs non-traveling exact solutions for higher-dimensional or variable-coefficient nonlinear evolution equations across diverse physical fields.