On symmetry solutions of nonlocal complex coupled dispersionless system using Darboux transformation and artificial neural networks

Abstract

The generalized coupled dispersionless equations describe the dynamics of a current-fed string in an external magnetic field. This study introduces a novel methodology leveraging the Darboux transformation to derive analytical solutions for these complex equations. The approach explores symmetry-preserving and non-preserving solutions, further refined using the Levenberg–Marquardt algorithm within a neural network framework. The network underwent thorough validation using relative $L_2$ errors during training and testing on clean and noisy data. During this validation, thorough tabular and graphical representations validated our analytical results, proving their reliability. We carefully analyzed the solution behaviors using various visualization techniques, such as contours, three-dimensional plots, and corresponding error graphs. This study comprehensively analyzes the system’s dynamics by integrating analytical methods with artificial neural networks, bridging theoretical predictions with empirical validations. The findings offer new insights into wave behavior, stability, and nonlinear interactions within the system, contributing significantly to mathematical physics.