Applications of Kumar Malik Method to $(3+1)$-Dimensional Wazwaz–Kaur Boussinesq Equation

The Wazwaz–Kaur equation is a significant nonlinear partial differential equation. This equation is used to describe nonlinear wave propagation in fields including fluid dynamics, plasma physics, and other areas of mathematical physics. In this study, we investigate analytical solutions of the $(3+1)$-dimensional Wazwaz–Kaur Boussinesq Equation (WKBE). These solutions play a significant role in describing complex wave structures and in understanding the underlying physical mechanisms. For the construction of exact analytical solutions, we utilize the Kumar–Malik Method (KMM), which offers a systematic procedure for addressing nonlinear partial differential equations. Using this approach, a wide class of exact solutions is obtained, including Jacobi elliptic solutions as well as their hyperbolic, trigonometric, and exponential limiting forms. All solutions are validated through symbolic computations performed in Maple Software. The obtained solutions are visualized through graphical representations to demonstrate how different physical parameters affect the wave profiles. These findings are of great use in understanding the nonlinear behavior of the $(3+1)$-dimensional WKBE and in extending it to a wider range of physical systems with multidimensional wave interactions. Overall, the results of this research confirm the effectiveness of the KMM and introduce new exact solutions that may provide a useful reference for future analytical and numerical studies of nonlinear evolution equations.