Solitonic solutions for the reduced Maxwell-Bloch equations via the Darboux transformation and artificial neural network in nonlinear wave dynamics

In this article, we explore nonlinear wave dynamics by presenting solitonic solutions for the reduced Maxwell-Bloch equations, a model relevant to both nonlinear optics and Bose-Einstein condensates. First, we write the Lax pair for the system and apply a Darboux transformation to obtain and analyze the soliton solutions. Then, we introduce a novel approach by integrating the Darboux transformation, a powerful analytical tool, with the Levenberg-Marquardt artificial neural network, a reliable numerical method. This combination enables the identification and validation of soliton solutions, supported by detailed graphical and tabular analyses. The Levenberg-Marquardt artificial neural network effectively demonstrates the uniqueness and convergence of the solutions, with the accuracy of the results validated through comprehensive graphical representations.