Relativistic effect on time-fractional KdV-soliton and its propagation in electron beam plasmas having two kappa-distributed electrons

In this work, we examine the nonlinear propagation of low-frequency ion-acoustic solitary waves in a collisionless, electropositive, relativistic electron-beam plasma composed of warm relativistic ions, superthermal cool and hot electron populations modeled by kappa distributions, and a finite-temperature relativistic electron beam. Starting from a relativistic multi-fluid description coupled with Poisson’s equation, we employ the reductive perturbation method to derive a planar Korteweg-de Vries (KdV) evolution equation that captures the interplay between quadratic nonlinearity and dispersion in this multi-component environment and then construct a time-fractional generalization of the KdV model via a variational formulation. The resulting time-fractional KdV equation is solved using the variational iteration technique, yielding first- and second-order analytical approximations that clearly display the impact of fractional temporal dynamics. Our parametric analysis shows that increases in the relativistic streaming factor, the hot-electron-to-ion density ratio, and the cool-electron spectral index reduce the linear phase velocity. In contrast, stronger cool-electron superthermality enhances the effective nonlinearity of the ion-acoustic mode. We further find that relativistic corrections substantially modify both the amplitude and width of classical solitons and their time-fractional counterparts: while standard KdV solitons maintain their form during propagation, fractional solitons undergo pronounced amplitude and structural changes reflecting the underlying memory effects. Overall, these findings outline a specific set of conditions in which relativistic movement, superthermal electron tails, and changes over time together influence how ion-acoustic solitary structures form, stay strong, and change in relativistic electron-beam plasmas.