Wave modelling of 3 + 1 dimensional Wazwaz Kaur Boussinesq equation with the bilinear neural network method.

This study investigates a newly developed (3 + 1)-dimensional Wazwaz-Kaur-Boussinesq equation, a significant advancement in fluid dynamics, nonlinear wave phenomena, and mathematical physics, due to its capability to model complex wave structures across multiple spatial and temporal dimensions. By applying the bilinear neural network method to a bilinearized form of the equation with the concept of corresponding tensor formula, we introduced new activation functions in a single-layer neural configuration using a "4-2-1" neuron structure. For the very first time we combined the bilinear neural network method with the (3 + 1)-dimensional Wazwaz-Kaur-Boussinesq equation. In our proposed method, the neural setup allows the network to efficiently capture and represent the nonlinear behaviour intrinsic to multidimensional wave propagation. Using Maple software for symbolic computation techniques, we systematically obtained lump, single, and periodic soliton wave solutions in the equation, revealing the complex dynamics of waves and unique aspects of the physical behaviour of the system. Such solutions and their parameters were investigated using complex 3D, 2D, contour, and density graphs to illustrate how parameter changes influence the form and dynamical features of the wave packets. Such artworks help better explain the dynamics of the exact solution of the equation and illustrate more the system's complex dynamics and physical features.