Fractional order effects on solitary waves and chaotic regimes in the mKdV Burgers equation.

This paper examines the space-time fractional modified Korteweg-de Vries Burgers (mKdV-Burgers) equation to address nonlinear wave dynamics of the equation through the improved F-expansion representation with the Riccati equation. The given strategy offers a methodical system of obtaining a wide category of precise analytical solutions, solitary wave solutions, kink-type solutions, periodic solutions, and rational solutions. The resulting results show the existence of dissipative and shock-like solitons, which add to the knowledge of nonlinear propagation phenomena in complicated media. Moreover, a dynamical study is performed in terms of bifurcation structures, phase portraits, Lyapunov exponents, and sensitivity analysis of changes between stable and chaotic states. These studies show that parameters of fractional order affect the stability and complexity of the system. This dynamical and analytical set of methods does not only confirm the efficiency of the improved F-expansion method but also contributes to new physical understanding of the fractional nonlinear evolution equations (FNLEEs). Findings may be generalized to fluid dynamics, plasma physics, and nonlinear optics, and the framework can be generalized to higher-dimensional or coupled fractional systems to control and predict multi-stable and chaotic behavior.