In this research, we present a novel nonlinear model of the extended \((2+1)\) -dimensional Calogero-Bogoyavlenskii-Schiff equation. This model builds upon the original \((2+1)\) -dimensional Calogero-Bogoyavlenskii-Schiff equation by incorporating a flux term, \(\Theta \mathcal {U}_{xy}\) , which characterizes the propagation of Riemann waves along the \(y\) -axis and long waves along the \(x\) -axis. Notably, the extended Calogero-Bogoyavlenskii-Schiff equation’s integrability is preserved with this flux term’s inclusion. The Intriguing Nature of Riemann waves holds immense significance in numerous fields, including the tumultuous tsunamis of rivers, the hidden internal waves of oceans, and the enchanting Magento-sound waves that resonate in plasmas. To unveil the mesmerizing traveling wave solutions, we employed a robust technique called the Generalized Arnous method, resulting in an array of solutions that encompass trigonometric, hyperbolic, and logarithmic functions. This remarkable technique can unveil a diverse spectrum of precise solutions, featuring vibrant bright solitons, shadowy dark ones, and elusive solitons. Through the application of the Galilean transformation, we reformulate the model into a planar dynamical system, wherein a qualitative investigation is undertaken. The bifurcation analysis of the planar dynamical system has been conducted utilizing bifurcation theory and phase portraits for dynamical analysis. The sets of bifurcation encompass the center, saddle point, and cuspidal point. Additionally, the chaotic and quasi-periodic patterns have been observed after introducing the perturbation term. The simulations show that adjusting the amplitude and frequency parameters can change the system’s dynamic behavior. Furthermore, quasi-periodic patterns are identified through chaos detection tools like Lyapunov exponent, and multi-stability analysis. Lastly, we conduct a rigorous sensitivity study to investigate how the system’s behavior changes in response to varying initial conditions. These findings offer novel contributions to studying the equations, greatly, enhancing our understanding of the dynamics in the nonlinear wave models.
