Abstract <p>This paper focuses on the (1+1)-dimensional geophysical Korteweg–de Vries equation to explain the complex behavior of nonlinear waves in different areas of mathematical physics, such as nonlinear optics, fluid dynamics, and plasma physics. Analytical solutions are obtained by employing two analytical techniques: the modified Khater method and the Sardar subequation technique. These techniques yield a variety of novel solutions for the system, which are systematically compared to enhance understanding of the underlying dynamics of the nonlinear model. The solutions include trigonometric, hyperbolic, rational, and Jacobi elliptic functions, providing a rich mathematical framework. Graphical simulations are presented to visualize the dynamical behavior of the obtained solutions, with 3D surface plots, 2D line graphs, and contour plots generated using software such as MATLAB and Mathematica. To further analyze the system’s qualitative behavior, phase portrait analysis is carried out for the unperturbed planar form. When an external forcing term is introduced, the system exhibits complex dynamics and chaotic behavior. This chaotic nature is demonstrated using time series, two- and three-dimensional phase plots, Poincaré maps, and the computation of Lyapunov exponents. Additionally, a comprehensive multistability analysis reveals the system’s high sensitivity to initial conditions, where small perturbations can induce transitions between stable and unstable regimes. Numerical simulations using the Runge-Kutta method support the analytical findings and highlight the intricate dynamical behavior of the model. In general, the analytical and numerical techniques employed offer valuable tools for exploring and understanding a wide range of non-linear wave phenomena.</p> Graphic abstract <p></p>