Novel analytical superposed nonlinear wave structures for the eighth-order (3+1)-dimensional Kac-Wakimoto equation using improved modified extended tanh function method

Higher-order nonlinear partial differential equations, such as the eighth-order Kac-Wakimoto model, are useful for studying wave turbulence in fluids, where energy transfers across a range of wave numbers. This phenomenon is observed in oceanographic research involving sea surface and internal waves, where intricate multi-dimensional interactions play a crucial role. In this work, we use the improved modified extended tanh function method for the first time to extract the exact solutions of the eighth-order (3+1)-dimensional Kac-Wakimoto equation, which describes the dynamics of fields and the structure of solutions in various physical and mathematical contexts. The proposed method is simple and quick to execute, and it offers more innovative solutions than other methods. As a consequence, through the donation of suitable assumptions for the parameters, some new solutions for dark and singular soliton, as well as Jacobi elliptic, exponential, hyperbolic, and singular periodic forms, are developed. Furthermore, to enhance understanding, graphical representations of certain solutions are included.