Exploring Exact Traveling Wave Solutions of Fractional Nonlinear Model Emerging from Surface Water Waves in a Channel: Bifurcation, Chaos, and Stability

In this work, we investigate the dynamic behavior of the time M-fractional regularized long wave Burger (tM-fRLW-Burger) equation, which describes surface water waves in a channel. The main focus of this research spans several key areas in engineering and mathematical physics. First, we perform a comprehensive bifurcation analysis of the tM-fRLW-Burger equation, methodically identifying equilibrium points and their stability characteristics under diverse parametric settings. We use both linear and nonlinear stability analysis methods to sort these equilibrium sites into groups based on their features, showing the differences between stable nodes, saddle points, and limit cycles. We demonstrate local and global stability through a comprehensive analysis of phase pictures, highlighting the impact of parameter variations on the system’s dynamic behavior. Second, we analyze the chaotic behavior of the system under different initial conditions and perturbation terms using tools such as two- and three-dimensional phase portraits, time series, Poincaré diagrams, and multi-stability. The incorporation of Poincaré maps offers a powerful tool for analyzing the topology of chaotic orbits, enabling a deeper understanding of the system’s underlying dynamical structure. Additionally, we originate the exact traveling wave for the tM-fRLW-Burger model employing the modified simple equation method. For specific values of the free parameters, we conduct a comprehensive analysis of the obtained solutions, accompanied by some unique diagrams. These solutions come in the form of trigonometric, exponential, and hyperbolic functions. For certain parameter values, we identify many wave phenomena, including kink waves, anti-kink waves, interactions between anti-kink and periodic waves, periodic lump and kink interactions, dark bell-shaped waves, and periodic waves, illustrating the model’s intricate solution structure. Finally, we observe the modulation instability of the proposed model.