Optical soliton stability in zig-zag optical lattices: comparative analysis through two analytical techniques and phase portraits

Abstract This article explores the examination of the widely employed zig-zag optical lattice model for cold bosonic atoms, which is commonly utilized to depict nonlinear wave in fluid mechanics and plasma physics. The focus is on obtaining soliton solutions in optics and investigating their physical properties. A wave transformation is initially applied to convert a partial differential equation (PDE) into an ordinary differential equation (ODE). Soliton solutions are subsequently obtained through the application of two distinct methods, namely the generalized logistic equation method and the Sardar sub-equation method. These solutions include bright, dark, combined dark-bright, chirped type solitons, bell-shaped, periodic, W-shape, and kink solitons. In this paper, the solutions derived from two analytical approaches were compared to enhance the understanding of the behavior of the discussed nonlinear model. The obtained solutions have significant implications across various fields such as plasma physics, fluid dynamics, optics, and communication technology. Furthermore, 3D and 2D graphs are generated to depict the physical phenomena of the derived solutions by assigning appropriate constant parameters. The qualitative evaluation of the undisturbed planar system involves the analysis of phase portraits within bifurcation theory. Subsequently, the introduction of an outward force is carried out to induce disruption, and chaotic phenomena are unveiled. The detection of chaotic trajectory in the perturbed system is achieved through 3D plots, 2D plots, time scale plots, and Lyapunov exponents. Furthermore, stability analysis of the examined model is addressed under distinct initial conditions. Finally, the sensitivity assessment of the model under consideration is carried out using the Runge–Kutta method. The results of this study are innovative and have not been previously investigated for the system under consideration. The results obtained underscore the reliability, simplicity, and effectiveness of these techniques in analyzing a variety of nonlinear models found in mathematical physics and engineering disciplines.