Nonlinear dynamics of dual acoustic waves in dusty plasmas on the lunar shady-side

Weakly nonlinear electrostatic waves are examined in a dusty-plasma model for the lunar nightside wake during the Moon's passage through the Earth's magnetotail. The linear spectrum contains a fast dust-ion-acoustic wave (DIAW), governed mainly by ion inertia, and a slow dust-acoustic wave (DAW), governed by dust inertia. A Korteweg--de Vries (KdV) reduction shows that the DAW remains rarefactive over the considered electron-depletion and temperature-ratio ranges, whereas the DIAW crosses a polarity-reversal boundary, where the quadratic nonlinearity vanishes. This boundary marks the limit of the KdV ordering rather than a physical amplitude singularity. Retaining cubic nonlinearity gives a Gardner description of the near-critical DIAW. Within the adopted lunar-wake parameter domain, the critical line remains in the positive-cubic-coefficient region, so the model admits polarity-selective Gardner solitons and the dual-polarity mKdV limit but does not reach the flat-top Gardner limiting sequence. The results identify how the two acoustic branches differ in nonlinear polarity and in the accessibility of near-critical electrostatic structures in this lunar-wake model.