Exploring Nonlinear Dynamics in Fluid Mechanics: Bifurcation, Chaos, and Soliton Analysis

ABSTRACT This paper reveals a nonlinear new dissipative Boussinesq model in a fluid mechanism system. This model is obtained from the quasi‐geostrophic barotropic potential velocity equation. We use a fractional derivative to improve accuracy as well as understand the dynamics of the model. It describes the nonlinear Rossby waves in the atmosphere. By utilizing the modified simplest equation method, we investigate the distinct kinds of exact wave solutions involving trigonometric, hyperbolic, and rational functions. To obtain and verify the solutions, we use the Mathematica software. We demonstrate the obtained solutions through two‐dimensional, three‐dimensional, and contour graphs by using the Mathematica tool. The main contribution of this paper is to analyze the qualitative analysis, including modulation instability, bifurcation analysis, chaotic behavior, and sensitivity nature. We explore the important characteristics of chaos through two‐dimensional (2‐D) phase portraits, three‐dimensional (3‐D) phase portraits, Poincaré sections, time series, and Lyapunov exponents. At the end, the gained solutions are fruitful in different areas, like oceanography, fluid dynamics, coastal engineering, and many others.