Nonlinear Dust‐Ion‐Acoustic Waves With Finite Dust‐Charge Relaxation: OML Susceptibility, Afterglow Validity, and Gardner–Burgers Shocks

ABSTRACT Finite dust‐charge relaxation provides a physically measurable memory mechanism in dusty plasmas, but its weakly nonlinear reduction must remain consistent with the charging law and with the frequency range in which the dust can be treated as immobile. We therefore reformulate the charging closure of a three‐component dust–ion‐acoustic model using orbital‐motion‐limited (OML) current balance rather than an imposed exponential equilibrium law. The OML floating‐potential relation is written explicitly through the Lambert function, and its first three local derivatives generate the susceptibilities entering a near‐critical reductive perturbation analysis. The resulting Gardner–Burgers equation retains a positive charge‐lag dissipation coefficient. Published radio‐frequency argon afterglow measurements are used only to anchor the existence and scale variation of finite charging relaxation; the late charge‐freeze‐out state is explicitly identified as lying outside the simultaneous short‐memory and immobile‐dust validity window. Ion–neutral collisions are incorporated as a weak‐damping extension and yield an additional linear attenuation term. Laboratory velocities are normalized by the ion‐acoustic speed, making the associated relativistic parameter of order for illustrative eV‐scale argon temperatures. The previous numerical scans at are removed from the afterglow study; all revised numerical results use , while the optional Lorentz term is retained only in the general coefficient formula for completeness. In the nonrelativistic branch, exact Gardner solitary waves are retained, while traveling Gardner–Burgers fronts are classified analytically by a discriminant separating monotone and oscillatory tails. Reproducible calculations illustrate the OML criticality, the shock threshold, and monotone energy decay. The revision therefore places the experimentally anchored, nonrelativistic finite‐charging model on a physically consistent footing while retaining the optional Lorentz correction only as a symbolic extension of the general coefficient formula.