Analysis of stability and chaotic trajectories in nonlinear fluid wave interactions under forcing effects

The Ablowitz–Kaup–Newell–Segur (AKNS) water wave equation is a fundamental model for describing nonlinear wave propagation in various physical systems, including fluid dynamics, plasma physics, and nonlinear optics. Its higher-dimensional form provides a more realistic framework for understanding complex wave interactions in dispersive media. In this work, the \((2+1)\) -dimensional AKNS equation is studied from both analytical and dynamical perspectives. Analytical solutions of the governing equation are obtained using the modified Khater method, constructing different types of wave structures such as stationary and traveling profiles. These solutions illustrate the influence of nonlinearity and dispersion on the formation and evolution of coherent wave patterns. To further explore the system behavior, a detailed dynamical analysis is performed. The model is reduced to a system of ordinary differential equations, allowing the investigation of phase space dynamics through phase portraits, time series, Poincaré sections, bifurcation diagrams, and Lyapunov exponents. The results demonstrate a wide range of behaviors, including periodic, quasi-periodic, and chaotic motions, depending on the choice of system parameters. In particular, the system shows strong sensitivity to initial conditions and parameter variations, indicating the presence of complex and chaotic dynamics.