Integrability, Soliton Dynamics, and Physics-Informed Neural Network Simulation of a Generalized Coupled KdV Equation

This paper studies the integrability and soliton dynamics of a generalized coupled Korteweg-de Vries (KdV) equation. The bilinear form of this equation is obtained through studying the connection between Bell polynomials and Hirota bilinear operators. The explicit analytical solutions, including one-soliton, two-soliton, and one-breather solutions are systematically constructed by using the perturbation expansion method, and their dynamical behaviors are illustrated through numerical simulations. Through long-time asymptotic analysis, the collision mechanism of two-soliton solutions is thoroughly examined. The results demonstrate that after collision, both solitary waves retain their original amplitudes and velocities. They experience only a phase shift, which confirms the completely elastic nature of soliton interactions in this coupled system. Physics-informed neural network (PINN) is employed to conduct numerical simulations for the soliton solutions, and the quantitative evaluation via relative \(L_{2}\) $L_2$ errors verifies the excellent consistency between the predicted and analytical solutions, which further validates the rationality of the derived analytical solutions. This finding is validated by both analytical derivations and numerical visualizations, providing compelling evidence for the system’s integrability and the purely elastic nature of soliton collisions.