Dynamics of N-soliton waves, lump-breathers, and M-lump collision with improved bilinear neural network method

Abstract

In this paper, we focus on the analytical study of the (3+1)-dimensional Geng equation by employing the improved bilinear neural network method which is quite effective in solving nonlinear equations. The Geng equation, which is used to describe the phenomena in the shallow water dynamics and nonlinear wave interactions, can be used to analyze the higher-dimensional wave system. We are thus able to construct a variety of solutions such as N-soliton waves, lump-breathers, and M-lump solutions under the condition of using multivariate test functions in the one-hidden-layer structure 4-3-1 and the two-hidden-layer structure 4-2-3-1 neural network model. These solutions capture intricate interactions and dynamic behaviors inherent to the Geng equation, offering deeper insights into its underlying structure. Additionally, in order to improve the interpretation of the obtained solutions, we offer a detailed graphical representation in the form of 3D, 2D and contour plots. The proposed methodology, thus, proves highest efficiency and reliability while solving high-dimensional nonlinear equations and opens the avenue for future study on physical and mathematical interpretation of such equations, especially in the context of shallow water wave modeling.