Nonlinear wave interactions and stability analysis in the CBS–nCBS model via bilinear neural network approach

The Calogero–Bogoyavlenskii–Schiff (CBS) equation serves as a cornerstone for nonlinear integrable partial differential systems that govern complex wave phenomena. This study investigates the [Formula: see text]-dimensional CBS-negative-order CBS (CBS–nCBS) model, unveiling novel soliton solutions. Using the bilinear neural network method, a bilinear representation of the model is derived, and multi-soliton solutions are identified through various neural network architectures. The bilinear neural framework employs single-layer architectures (4–3–1, 4–4–1) combined with diverse test functions to capture kink–breather waves, M-lumps, and lump–kink interaction solitons. A novel Jacobi elliptic activation function is applied to explore rogue wave soliton solutions. Spatiotemporal visualizations, such as 3D with contour, density, and 2D plots, generated using MATLAB for selected parameters, offer valuable insights into the structural evolution of these solutions. Moreover, the stability analysis of the governing equation is investigated using linear stability analysis under small perturbations, and the results are illustrated with dispersion analysis graphs. These visualizations offer a quantitative representation of the intricate dynamics at play. The bilinear neural methodology, utilizing established test functions, provides a systematic and robust approach for analyzing complex nonlinear phenomena. The findings enhance the understanding of soliton structures and expand the applicability of the CBS–nCBS equation in nonlinear wave theory.