Dynamics of soliton solutions, bifurcation, chaotic behavior, stability, and sensitivity analysis of the time-fractional Gardner equation

In the present article, we investigate the stability, chaos, and bifurcation analysis of an eminent nonlinear evolution equation, namely, the time-fractional Gardner equation, which is a hybrid model that includes properties of both the Korteweg–de Vries (KdV) and modified KdV equations. Exact analytical solutions are established using the improved F-expansion approach, which is a powerful tool for generating soliton and periodic wave solutions. By means of suitable transformations, the equation is transformed into a further manageable form, and then, the analytical soliton solutions, such as the bell-shaped, concave, convex, kink, anti-kink, breather, and lump-shaped solitons, are investigated. The dynamical behaviors of these solutions are further explored through bifurcation analysis, sensitivity to initial conditions, and the detection of chaos. We conduct both linear and Hamiltonian approaches to examine the stability and validity of the physical significance of the solutions. Graphical outset and phase portraits support the analytical results, revealing the transition from regular to chaotic behavior under parameter variations. The results provide new insights into nonlinear wave propagation and provide a foundation for future research in complex physical systems, such as optics, plasma, and fluid dynamics.