Periodic–Kink, Lump, and Rogue Waves with Parameter Discovery in an Integrable Model: Bilinear and Physics-Informed Neural-Network Approaches

This research investigates nonlinear wave dynamics of the integrable (3+1)-dimensional extended Bogoyavlensky- Konopelchenko equation, which models the interaction of a Riemann wave and a long wave in fluid dynamics. The equation has important applications in the modeling of shallow water waves, ion acoustic waves, and the propagation of fluids. Initially, we employ Hirota’s bilinear framework with symbolic single-hidden-layer neural-network architectures, referred to as bilinear neural networks (BNNs), and prescribed analytical activation functions to construct complex wave structures, including periodic–kink waves, M-lumps, lump–kink interactions, and modulated lump solutions. A Jacobielliptic activation function is employed within the symbolic BNN ansatz to construct and investigate rogue-wave solutions. In the second phase, an enhanced Physics-Informed Neural Network (PINN), equipped with a global adaptive activation function and residual-based attention, is introduced for physics-informed solution approximation and inverse parameter identification. The proposed PINN accurately reproduces the periodic–kink, M-lump, and lump–kink dynamics, showing good agreement with the analytical solutions and low error metrics, while the comparative analysis highlights the complementary strengths and limitations of the analytical and physics-informed numerical approaches. The framework also exhibits good stability in recovering nonlinear coefficients from noisy data, demonstrating its potential for parameter discovery. The results demonstrate that combining BNN-based exact solution construction with enhanced PINN-based numerical forward and inverse modeling provides a versatile framework for nonlinear wave analysis and the investigation of complex wave phenomena in applied mathematics and nonlinear physics.