Chaos, Bifurcation, and Soliton Stability of Beta-Fractional Nonlinear Model for Cold Bosonic Atoms

In this study, we investigate the wave dynamics of nonlinear phenomena arising in zig-zag optical lattices of cold bosonic atoms using a beta fractional nonlinear model. The aim of this article is to examine its chaotic properties, bifurcation, nonlinear dynamics, and soliton behaviors. Using the analytical expansion method, we derive a variety of exact soliton solutions, including hyperbolic, trigonometric, and exponential functions, which yield periodic, bell-shaped, anti-bell-shaped, breather, and lump solitons. We also examine the effect of the fractional-order derivative on soliton dynamics and stability, supported by three- and two-dimensional diagrams of obtained solitons. Furthermore, this study investigates the modulation instability using the linear stability approach and the formation of chaotic and quasi-periodic responses under perturbed external periodic forcing. Phase-plane trajectories are used to analyze bifurcation, chaos, and quasi-periodic dynamics. These results provide new viewpoints for applications in nonlinear optics and quantum computing while enriching the understanding of soliton phenomena in optical lattices. The findings of this research support the reliability of the employed methods in further exploration of nonlinear evolution equations.