Exploring soliton dynamics in Zakharov equations: Analytical approach and LM-ANN-driven convergence of solitary waves

Abstract

This study aims to derive soliton solutions for the Zakharov equations, a versatile model that generalizes several significant phenomena in science and engineering. The equations governing laser-plasma interactions are formulated, resulting in a system that accommodates complex soliton structures. First, the system is reformulated into nonlinear ordinary differential equations using a traveling wave transformation. Exact analytical solutions are obtained using the Kumar–Malik (KM) method, yielding multiple classes of soliton solutions expressed in terms of hyperbolic, Jacobi elliptic, trigonometric, and exponential functions, subject to appropriate parameter constraints. These solutions enable the construction of a wide variety of soliton waveforms, including periodic, dark, bright, and W-shaped soliton waves. Secondly, to assess the stability and accuracy of these solutions, the Levenberg–Marquardt artificial neural network (LM-ANN) approach is employed. This method demonstrates high reliability in approximating the analytical profiles, with convergence and precision confirmed through fitness and regression analyses. Furthermore, the study includes extensive 3D, 2D, and density plots of the obtained solutions to facilitate a deeper understanding of the soliton dynamics. This research advances the study of soliton dynamics in nonlinear systems and introduces a novel integration of neural network methodologies with classical soliton theory, with potential applications in optical communication, ultrafast photonics, and nonlinear signal processing.