The seminal paper of Mamun et al. [Phys. Plasmas 3, 702 (1996)] is revisited within the theoretical framework of the Tsallis statistical mechanics. The nonextensivity may originate from the correlation or long-range interactions in the dusty plasma. It is found that depending on whether the nonextensive parameter q is positive or negative, the dust-acoustic (DA) soliton exhibits compression for q<0 and rarefaction for q>0. The lower limit of the Mach number for the existence of DA solitary waves is greater (smaller) than its Maxwellian counterpart in the case of superextensivity (subextensivity).

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The head-on collision between two ion-acoustic solitons (IASs) is studied in pair ions plasmas with hybrid Cairns–Tsallis-distributed electrons. The chosen model is inspired from the experimental studies of Ichiki et al. [Phys. Plasmas 8, 4275 (2001)]. The extended Poincaré–Lighthill–Kuo (PLK) method is employed to obtain the phase shift due to the IASs collision. Both analytical and numerical results reveal that the magnitude of the phase shift is significantly affected by the nonthermal and nonextensive parameters (α and q), the number density ratios (μ and υ) as well as the mass ratio σ. For a given mass ratio σ ≃ 0.27 $\sigma \simeq 0.27$ (Ar+, SF 6 − ${\text{SF}}_{6}^{-}$ ), the magnitude of the phase shift Δ Q ( 0 ) ${\Delta}{Q}^{\left(0\right)}$ decreases slightly (increases) with the increase of q (α). The effect of α on Δ Q ( 0 ) ${\Delta}{Q}^{\left(0\right)}$ is more noticeable in the superextensive distribution case (q < 1). As σ increases [ σ ≃ 0.89 $\sigma \simeq 0.89$ (Xe+, SF 6 − ${\text{SF}}_{6}^{-}$ )], the phase shift becomes wider. In other terms, the phase shift was found to be larger under the effect of higher densities of the negative ions. Our findings should be useful for understanding the dynamics of IA solitons’ head-on collision in space environments [namely, D-regions ( H + ${\text{H}}^{+}$ , O 2 − ${\text{O}}_{2}^{-}$ ) and F-regions (H+, H−) of the Earth’s ionosphere] and in laboratory double pair plasmas [namely, fullerene (C+, C−) and laboratory experiment (Ar+, F−)].