Ion-acoustic waves in dense quantum plasmas are strongly influenced by Fermi degeneracy, Landau quantization, and finite-temperature effects, and in many relevant environments, they also experience memory and nonlocal transport processes that cannot be captured within the planar integer Korteweg-de Vries (KdV) paradigm. In the present work, we revisit this problem by considering a two-fluid, partially degenerate electron-ion plasma in which electron trapping in the presence of a quantizing field and finite temperature is taken into account. Starting from the normalized fluid-Poisson system appropriate for such magnetized quantum plasmas, the reductive perturbation technique is used to derive the planar integer KdV equation for weakly nonlinear ion-acoustic disturbances. Within this integer-order KdV framework, we recast the evolution equation as a planar dynamical system, construct the associated Hamiltonian and effective Sagdeev-like potential, and demonstrate the existence of compressive solitary waves and nonlinear periodic modes via homoclinic and periodic phase-space orbits. Exact multi-soliton solutions and interaction states are then obtained by combining Hirota’s direct bilinear method with generalized Wronskian representations, allowing us to describe not only standard one-, two-, and three-soliton profiles but also positon-negaton interactions relevant to magnetized, partially degenerate plasmas. To incorporate hereditary and history-dependent effects that arise from anomalous transport and nonlocal temporal response in dense environments, we extend the model by introducing a Caputo time-fractional derivative, thereby obtaining a time-fractional KdV (FKdV) equation that continuously connects the classical KdV limit to fractional dynamics. The FKdV equation is analyzed using the Tantawy technique. This semi-analytical iterative scheme yields rapidly convergent series approximations for the fractional ion-acoustic soliton and provides explicit control of the approximation error. The fractional solutions show that varying the order of the Caputo derivative modifies the amplitude, width, and temporal relaxation of the solitary structures and can even split the pulse into two distinct lobes, in contrast with the nearly rigid propagation predicted by the integer-order KdV equation. Taken together, these results clarify how Landau quantization, finite electron temperature, and fractional-order memory jointly shape the morphology, robustness, and interaction properties of ion-acoustic structures in strongly magnetized quantum plasmas of astrophysical and high-energy-density laboratory interest.
Alzaben et al. (Fri,) studied this question.