Physics-informed neural networks with energy constraints for coupled KdV equations: analytical and computational insights into soliton interactions

Abstract

Nonlinear wave phenomena play a central role in fluid dynamics and plasma physics. Soliton interaction is a key feature of such systems. The coupled Korteweg-de Vries (KdV) equations describe these nonlinear and dispersive waves. However, obtaining accurate and stable solutions remains difficult. This study presents a combined analytical and machine-learning approach. Analytical two-soliton solutions are derived using Hirota's bilinear method. Dispersion relations and interaction properties are obtained to explain the evolution of wave profiles. Interaction times and peak positions are computed numerically to track soliton behavior before, during, and after collision. A modified physics-informed neural network (PINN) is then developed. Global physical constraints, such as energy conservation, are included directly in the loss function. This ensures that the learned solutions remain consistent with the underlying physics. Numerical experiments show that the PINN framework provides improved stability and accuracy. The model captures long-term nonlinear wave interactions with higher fidelity. The proposed method offers a reliable computational framework for analyzing coupled KdV systems and other nonlinear dispersive wave models.