Bifurcation Analysis, Chaotic Behavior, Sensitivity Analysis, Stability Analysis, and Exploring Exact Traveling Wave Solutions of M‐Fractional Nonlinear Model

ABSTRACT In this article, we examine the dynamical behavior of the time M‐fractional modified regularized long wave Burger (tM‐fMRLW‐Burger) equation, which models surface water waves in a channel. First, we perform a bifurcation analysis of the proposed model to systematically identify equilibrium points. By varying the parameters, the local and global behavior of the system is illustrated through the phase portraits. Second, we explore the system's chaotic behavior under different initial conditions through phase portraits in 2D and 3D, time series, Poincaré diagrams, and multistability analysis. We also derive exact traveling wave solutions for the modified RLW‐Burger model using the modified simple equation method. For specific values of the free parameters, we provide a detailed analysis of these solutions, supported by unique diagrams. Furthermore, the modulation instability of the proposed model is investigated in detail. Understanding the dynamic properties of such systems is essential for predicting outcomes and developing new technologies. The study enhances comprehension of intricate hydrodynamic processes, facilitating innovative solutions for water management and the development of resilient infrastructure to sustainably tackle water‐related concerns.