Propagation of nonlinear dynamics in atomic chains using chaos theory

This article presents a variety of novel soliton solutions and a comprehensive dynamical analysis for a complex nonlinear equation modeling the behavior of nonlinear atomic chains with long-range interactions. The governing model is first reduced to lower dimensions via a wave transformation. Utilizing the extended algebraic method, multiple soliton solutions such as kink, double-periodic wave solitons, anti-kink, and dark solitons are systematically derived. These solutions are then illustrated through detailed visualizations, including 2D and 3D graphical representations, to enhance the physical interpretation. Subsequently, a phase portrait analysis is carried out for the critical points of the unperturbed system. We also classify the phase portraits using the Hamiltonian function to further investigate the system’s qualitative behavior. Upon introducing an external perturbation, the resulting dynamical behavior is investigated using advanced chaos detection techniques. These include 2D and 3D phase portraits, Poincaré sections, time series analysis, multistability behavior, quasi-periodic attractors, fractal dimension, bifurcation diagrams, power spectra, and Lyapunov exponents. The soliton solutions have applications in plasma physics, fluid dynamics, nonlinear optics, and communication systems.