Exploring chaotic behavior, conservation laws, Lie symmetry, and soliton dynamics in the generalized $A\mp$ equation

In this study, the generalized $A\mp$ equation is explored, with symmetry generators addressing the criteria for Lie invariance. The proposed approach yields the Lie algebra, where translation symmetries in space and time correspond to mass conservation and energy conservation, respectively. By employing Lie group methods, the generalized $A\mp$ equation is transformed through suitable similarity transformations into a system of highly nonlinear ordinary differential equations. The modified F-expansion approach is then applied to derive soliton solutions. The behavior of these solutions is visualized in three and two dimensions (3D and 2D), with contour plots, and the effect of wave speed is studied for specific values of the physical components in the equation. These results contribute significantly to advancing the field by enhancing the depth and impact of research. Subsequently, the dynamic behavior of the model was thoroughly investigated, with particular emphasis on the chaos analysis. The incorporation of an external periodic force led to the emergence of chaotic and quasi-periodic phenomena. These complex dynamics were illustrated using time series plots, 2D and 3D phase portraits, return maps, bifurcation diagrams, chaotic attractors, fractal dimensions, Poincaré maps, and Lyapunov exponent analysis. This comprehensive approach not only provides deeper insight into the system's stability and sensitivity but also offers a valuable framework for identifying and controlling complex behaviors in nonlinear dynamic models.