Fractional Memory Effects in Dust-Acoustic Solitons: Multi-Soliton Dynamics and Analytical Advances for Lunar Terminator Plasma—Part (I), Planar Analysis

In this investigation, the nonlinear dust-acoustic waves in the lunar terminator region are studied in a three-component complex plasma comprising Boltzmann-distributed electrons and ions and inertial, cold, negatively charged dust grains. The fluid model is reduced, via the reductive perturbation technique, to a planar Korteweg–de Vries (KdV) equation that governs the evolution of small-amplitude dust-acoustic structures in this environment. Hirota’s direct method is then employed to derive exact multiple-soliton solutions, which allow us to examine the parameter dependence of dust-acoustic solitons and to characterize their overtaking collisions. The analysis shows that the soliton polarity and amplitude are controlled by the equilibrium electron–ion density ratio and the electron-to-ion temperature ratio, and that multi-soliton interactions remain elastic, with only finite phase shifts after collision. In the second part of the study, the planar integer KdV model is generalized to a time-fractional KdV (FKdV) equation to incorporate nonlocal temporal memory effects in the dust-acoustic dynamics. This FKdV equation is analyzed using two analytical approximation schemes: the Tantawy technique, recently proposed as a direct and rapidly convergent approach to fractional evolution equations, and the new iterative method, a widely used high-accuracy scheme in the fractional literature. For both methods, higher-order approximations are constructed, and their absolute and global maximum residual errors are quantified. The results demonstrate that the Tantawy technique provides compact approximations with superior accuracy and stability compared with the new iterative method for the present FKdV-soliton problem. The combined integer- and fractional-analytic framework provides a physically transparent framework for understanding how nonlinearity, dispersion, and fractional memory jointly shape dust-acoustic solitary structures in the electrostatically complex lunar terminator plasma, which is of paramount interest for future lunar missions like Luna-25 and Luna-27.