This article investigated the solitary wave solutions to the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries equation. The proposed model is particularly applicable to shallow water wave dynamics and may also extend to contexts such as acoustic wave propagation, nonlinear electric media, and oceanic wave phenomena. First, we constructed the ordinary differential equation form of the nonlinear partial differential equation with the help of the traveling wave transformation. After that, we utilized the generalized Arnous method and the modified sub-equation method to construct the solitary waves containing hyperbolic, exponential, trigonometric, and inverse functions. Using suitable parameter values, the graphical aspects of solutions are demonstrated by plotting a 3D surface plot (including a contour and density plot), a 2D surface plot, a streamline plot, and a polar plot. By utilizing these approaches, accurate analytical solutions for soliton waves were generated, which comprise kink, bright, and dark waves. We employed the generalized Arnous method and the modified sub-equation method to formulate a technique for addressing integrable systems, providing a valuable framework for examining nonlinear phenomena across various physical contexts. This study's outcomes enhance both nonlinear dynamical processes and solitary wave theory.

The complex dynamics of a slow-fast predator-prey interaction with herd behavior are examined in this work. We investigate the presence and stability of fixed points. By employing the bifurcation theory, it is shown that the model undergoes both a period-doubling and a Neimark-Sacker bifurcation at the interior fixed point. Under the influence of period-doubling and Neimark-Sacker bifurcations, chaos is controlled using the hybrid control approach. Moreover, numerical simulations are carried out to highlight the model's complexity and show how well they agree with analytical findings. Employing the slow-fast factor as the bifurcation parameter shows that the model goes through a Neimark-Sacker bifurcation for greater values of the slow-fast factor at the interior fixed point. This makes sense because if the slow-fast factor is large, the growth rates of the predator and its prey will be about identical, automatically causing the interior fixed point to become unstable owing to the predator's slow growth.

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.