The effects of variable dust size and charge on dust acoustic waves propagating in a hybrid Cairns–Tsallis complex plasma

Abstract

The propagation characteristics of dust acoustic waves (DAWs) in a dusty plasma consisting of variable size dust grains, hybrid Cairns–Tsallis-distributed electrons, and nonthermal ions are studied. The charging of the dust grains is described by the orbital-motion-limited theory and the size of the dust grains obeys the power law dust size distribution. To describe the nonlinear propagation of the DAWs, a Zakharov–Kuznetsov equation is derived using a reductive perturbation method. It is found that the nonthermal and nonextensive parameters influence the main properties of DAWs. Moreover, our results reveal that the rarefactive waves can propagate mainly in the proposed plasma model while compressive waves can be detected for a very small range of the distribution parameters of plasma species, and the DAWs are faster and wider for smaller size dust grains. Applications of the present results to dusty plasma observations are briefly discussed.

Appendix: Expressions in the dust charging process

Here, we present the used expressions in calculating the dust charging process

$$\begin{aligned} C_{1} & = - \frac{{A_{q\alpha } \left( {15 - 8\alpha_{i} + s\left( {6\alpha_{i} - 15} \right)\psi_{j0} } \right) + 15\sigma \gamma_{1} \left( {4q - 3} \right)\left[ {1 + \left( {q - 1} \right)s\sigma \psi_{j0} } \right]^{{\frac{q}{q - 1}}} }}{{\psi_{j0} \left\{ {3A_{q\alpha } \left( {5 + 8\alpha_{i} } \right) + 45\sigma \gamma_{0} \left( {4q - 3} \right)\left[ {1 + \left( {q - 1} \right)s\sigma \psi_{j0} } \right]^{{\frac{q}{q - 1}}} } \right\}}} \\ C_{2} & = \frac{{s\gamma_{2} - 15s\sigma^{2} \left( {\gamma_{3} + \gamma_{4} + \gamma_{5} } \right)\left( {3q - 2} \right)\left( {4q - 3} \right)\left[ {1 + \left( {q - 1} \right)s\sigma \psi_{j0} } \right]^{{\frac{1}{q - 1}}} }}{{2\psi_{j0} \left\{ {3A_{q\alpha } \left( {5 + 8\alpha_{i} } \right) + 45\sigma \gamma_{0} \left( {4q - 3} \right)\left[ {1 + \left( {q - 1} \right)s\sigma \psi_{j0} } \right]^{{\frac{q}{q - 1}}} } \right\}}} \\ \end{aligned}$$

where

$$\begin{aligned} \gamma_{0} & = 10 + 8\alpha_{e} + q\left\{ {q\left( {8s^{2} \alpha_{e} \sigma^{2} \psi_{j0}^{2} + 30} \right) - 4s\alpha_{e} \sigma \psi_{j0} \left( {s\sigma \psi_{j0} + 2} \right) - 35} \right\} \\ \gamma_{1} & = 30 - 105q + 90q^{2} - 8\alpha e + 8\left( {4 - 5q} \right)s\alpha_{e} \sigma \psi_{j0} + 4q\left( {10q - 7} \right)s^{2} \alpha_{e} \sigma^{2} \psi_{j0}^{2} \\ \gamma_{2} & = A_{q\alpha } \left\{ {15\left( {1 + 2C_{1} \psi_{j0} - s\psi_{j0} } \right) - 32\alpha_{i} \left( {C_{1} \psi_{j0} + 2s\psi_{j0} - 1} \right)} \right\} \\ \gamma_{3} & = 15q\left( {2q - 1} \right) + 32\alpha_{e} + 2\left\{ {C_{1} \left( {15q\left( {2q - 1} \right) - 16\alpha_{e} } \right) + 16\left( {4q - 3} \right)s\alpha_{e} \sigma } \right\}\psi_{j0} \\ \gamma_{4} & = \left\{ {15C_{1}^{2} q\left( {2q - 1} \right) + 32C_{1} s\alpha_{e} \sigma + 4\left( {16 + q\left( {30q - 43} \right)} \right)s^{2} \alpha_{e} \sigma^{2} } \right\}\psi_{j0}^{2} \\ \gamma_{5} & = 8C_{1} q\left( {10q - 7} \right)s^{2} \alpha_{e} \sigma^{2} \psi_{j0}^{3} + 12C_{1}^{2} q\left( {2q - 1} \right)s^{2} \alpha_{e} \sigma^{2} \psi_{j0}^{4} . \\ \end{aligned}$$