Symbolic computation meets bilinear neural networks: solving Caloero-Bogoavlenskii-Schiff equation for various wave solutions

This paper innovatively constructs a novel test function system based on generalized activation functions, successfully addressing the challenge of multi-layer membrane structure solutions for the Calogero-Bogoyavlenskii-Schiff (CBS) equation through the bilinear neural network method (BNNM). Within the classical framework of the bilinear approach, this study creatively introduces the AiryAi function into the construction of auxiliary functions for the first time, achieving its mathematical integration with the JacobiCN elliptic function and power functions to form a highly nonlinear hybrid function space. Leveraging the powerful symbolic computation capabilities of the Maple platform, we not only obtain exact analytical solutions ranging from single-layer to four-layer membrane structures, but also reveal the underlying mechanisms of complex dynamical behaviors such as Airy-type lump waves, multi-wave interactions, and rogue waves. The research not only enriches the theoretical framework of multi-soliton structures in nonlinear physics, but also opens up new directions in mathematical modeling and intelligent solution methods for complex systems.