Lump solitions, fractal soliton solutions, superposed periodic wave solutions and bright-dark soliton solutions of the generalized (3+1)-dimensional KP equation via BNNM

In this study, the bilinear neural network method (BNNM) is employed for seeking analytical solutions for the generalized (3+1)-dimensional KP equation, which are subtly constructed by both single-layer “4-3-1” and double-layer “4-2-2-1” neural network architectures. By constructing different activation functions and using Maple software for calculations, we obtained a large number of precise analytical solutions. After conducting a series of experimental assignments, we chose some appropriate parameters. These were then substituted into analytical solutions to highlight the final results more effectively and ensure they comply with physical laws. Ultimately, we obtained lump solutions, fractal soliton solutions, superposed periodic wave solutions, and bright-dark soliton solutions. The dynamic characteristics of these solutions are visualized using three-dimensional graphics, curve plots, density maps, and contour diagrams. These results offer valuable insights into nonlinear phenomena across diverse fields such as optics, acoustics, heat transfer, fluid dynamics, and classical mechanics. At the same time, we have applied BNNM to the generalized (3+1)-dimensional KP equation for the first time and demonstrated its effectiveness. Compared to traditional methods, BNNM exhibits significant advantages, which suggests it will be increasingly utilized in various types of nonlinear research.