A normalized time-fractional Korteweg–de Vries equation

A novel normalized time-fractional Korteweg–de Vries (KdV) equation is presented to investigate the effects of fractional time derivatives on nonlinear wave dynamics. The classical KdV model is extended by incorporating a fractional-order derivative, which captures memory and inherited properties in the evolution of soliton-like structures. Computational studies of the equation's nonlinear dynamics use a numerical scheme designed for the fractional temporal dimension. Simulations show that as the fractional parameter α decreases from 1 (the classical case) to smaller values, soliton dynamics change significantly. The soliton amplitude decreases, and its width increases. These changes are interpreted as dispersive or dissipative effects introduced by the fractional time component. At lower values of α, the soliton becomes broader and flatter, and its propagation is slowed. At intermediate values of α, multiple peaks and broader waveforms are observed, which implies more complex nonlinear interactions under fractional time evolution. The importance of fractional time derivatives in modifying the behavior of soliton solutions is highlighted, which demonstrates their potential in modeling physical systems where memory effects play a crucial role. The computational results provide insights into fractional partial differential equations and create new opportunities for future research in nonlinear wave propagation under fractional dynamics.