WEIFAN HOU, Na Qie, Ji‐Huan He, Jinfeng Ma, MEIGENG GAO, Zhou Chen, Chun‐Hui He
The fractional-order modified Korteweg–de Vries (mKdV) equation serves as a foundational model for the description of complex nonlinear wave phenomena in fluid mechanics, plasma physics, and associated disciplines. This equation is distinguished by its capacity to capture nonlocal interactions and memory effects through the utilization of fractional derivatives. This study utilizes the exponential function method as a standalone analytical technique to systematically derive a diverse set of exact solutions for the fractional-order mKdV equation. A thoroughgoing fractional complex transformation of the original partial differential equation results in the reduction of said equation to an ordinary differential equation. This differential equation is then solved by assuming a solution form composed of exponential functions. This methodological approach yielded five distinct families of exact solutions, validated through solitary wave profile simulations and consistency checks against the physical interpretability of fractional-order parameters. The obtained solutions provide analytical benchmarks for validating numerical algorithms in fractional-order systems and offer insights into nonlocal physical processes, such as long-range Coulomb interactions in plasmas or wave propagation in porous media with fractal structures. This work demonstrates the efficacy of the exponential function method in accessing the extensive solution space of fractional nonlinear equations, thereby establishing a foundation for its broader implementation in models of complex dynamics across scientific and engineering domains.