Analyzing Bifurcation Phenomena, Quasiperiodic Patterns, and Wave Structures of the Time-Fractional Benjamin–Bona–Mahony–Burger Equation

The Time-Fractional Benjamin–Bona–Mahony–Burger (TF-BBMB) equation is crucial for elucidating the one-way movement of long waves within dispersive and dissipative media, making it essential in modeling physical phenomena such as shallow water waves, plasma dynamics, and fluid transport processes. First, using the [Formula: see text]-expansion method, we investigate the soliton solutions of the TF-BBMB equation. The method parameterizes new soliton solutions into three cases, being hyperbolic-type, trigonometric-type, and rational-type functions. The paper also explores the representations of 3D graphs for these solutions for four different values of the fractional variable, [Formula: see text], and analyzes the impact of [Formula: see text] on each solution. In addition to constructing soliton solutions, we perform a bifurcation analysis and examine the chaotic dynamics of the TF-BBMB equation. By reducing the equation into a corresponding dynamical system, we explore its phase portraits, fixed points, and the conditions for bifurcations. This approach provides a comprehensive understanding of the qualitative behavior of the system, including the emergence of chaotic trajectories under specific parametric conditions, with the help of Lyapunov exponent analysis and Poincaré section.