On the time-fractional kinetic Alfvén solitary and cnoidal waves in a low-beta magnetoplasma in Saturn’s environment

In this work, the propagation of kinetic Alfvén solitary waves (KASWs) and kinetic Alfvén cnoidal waves (KACWs) is investigated in a strongly magnetized, low- β electron-ion plasma with ion pressure anisotropy and superthermal electrons, motivated by conditions in Saturn’s magnetospheric environment. Starting from a two-fluid description with a κ -distributed electron population and Chew-Goldberger-Low (CGL)-type ion pressure anisotropy, the reductive perturbation method is used to derive a planar Korteweg-de Vries (KdV) equation governing small-amplitude kinetic Alfvén structures. The model is then generalized to a time-fractional KdV equation by replacing the first-order temporal derivative with a Caputo fractional derivative to incorporate temporal nonlocality and memory effects in the plasma response. The resulting fractional evolution problem is treated using the Tantawy technique, which yields rapidly convergent analytical approximations for both fractional KASWs and fractional KACWs. The analysis shows that, for the low- β anisotropic conditions considered, only compressive kinetic Alfvén structures are supported in the sub-Alfvénic regime, and that superthermality, ion pressure anisotropy, plasma beta, and propagation obliqueness jointly control the amplitudes and widths of the solitary and cnoidal structures. Increasing the superthermal index reduces the deviation from Maxwellian behavior and produces stronger, broader compressive pulses, while larger ion pressure anisotropy and plasma beta mainly broaden the profiles with only minor changes in amplitude. The fractional order modulates the temporal development and localization of the nonlinear structures: reducing the order weakens the peak amplitude and slows the formation of sharp kinetic Alfvén fronts, reflecting the enhanced memory inherent to fractional dynamics. These results may assist in the interpretation of kinetic Alfvén activity in low- β space and astrophysical plasmas with ion pressure anisotropy and superthermal particles, such as Saturn’s magnetosphere and the near-Earth magnetosheath.