New Solitary Solutions and Bifurcation Analysis for the Space-Time Fractional mBBM Equation Using a Reliable Approach

The exact traveling-wave solutions, equilibrium structures, stability characteristics, bifurcation phenomena, and complicated nonlinear dynamics of a space-time fractional modified Benjamin–Bona–Mahony (mBBM) equation are thoroughly discussed in this article. The governing fractional partial differential equation is converted to a nonlinear ordinary differential equation by implementing a suitable traveling-wave transformation, wherein the fractional-order parameter β serves as a spatial–temporal scaling factor in the physical coordinate frame. This ordinary differential equation is then solved using the improved modified extended tanh-function approach. Various explicit traveling-wave solutions are successfully constructed and rigorously classified into smooth localized solitary waves (such as bell-shaped and kink-type profiles), doubly periodic Jacobi elliptic functions, singular periodic waves, and rational solutions. Their explicit domains and singularities are clearly discussed. Eigenvalue investigation and linearization are employed to identify the equilibrium points and evaluate their local stability. Furthermore, changes between regular and complex patterns of behavior, such as periodic, quasi-periodic, and chaotic dynamics, are shown via bifurcation and numerical analysis by introducing an external periodic excitation into the reduced planar system. The findings reveal that the fractional parameter β modifies the wave profile geometry and propagation speed, while the chaos is mainly influenced by external forcing parameters and nonlinear system coefficients. The improved modified extended tanh-function approach, along with dynamical-systems analysis, offers a comprehensive foundation for expressing both explicit wave patterns and complicated nonlinear processes.