## 2 Theoretical Model Equations under Plasma Assumption  
An unmagnetized collisionless plasma by composing of inertial pair-ion (e.g., positive ion (\(K^{+}\)) with mass (\(m_{pi}\)) and temperature (\(T_{pi}\)), and negative ion (\(SF_{6}^{-}\)) with mass (\(m_{ni}\)) and temperature (\(T_{ini}\))) and inertial-less (\(\alpha,q\)) distributed electrons, where \(\alpha\) and \(q\) are treated as the population of non-thermal and the strength of non-extensivity electrons, respectively, is considered. Under the above plasma assumption, one obtains the charge neutrality condition as \(1=N_{r1}+N_{r2}(N_{r1}=N_{ni0}/N_{pi0},N_{r2}=N_{e0}/N_{pi0})\), where \(N_{pi0}\), \(N_{ni0}\), and \(N_{e0}\) are the unperturbed densities of positive ions, negative ions, and electrons, respectively. In addition, the density function for \((\alpha,q)\)-distributed electrons is defined [25] as  
\[\rho_{e}=\rho_{e0}\left[1+(q-1)\frac{e\phi}{k_{B}T_{e}}\right]^{\frac{(q+1)}{ 2(q-1)}}\times\left[1+B_{1}\left(\frac{e\phi}{k_{B}T_{e}}\right)+B_{2}\left( \frac{e\phi}{k_{B}T_{e}}\right)^{2}\right], \tag{1}\]  
where \(B_{1}=-16q\alpha/\left(3-14q+15q^{2}+12\alpha\right)\), \(B_{2}=-B_{1}\left(2q-1\right)\), \(T_{e}\) is the electron temperature, \(k_{B}\) is Boltzmann constant, and \(\phi\) is the electrostatic potential function. It is noted that the above expression is recovered by (i) the Maxwell-Boltzmann distributed concentration of electrons when \(q\to 1\) and \(\alpha=0\), (ii) the cairn distributed concentration of electrons when \(q\to 1\) and \(\alpha\
eq 0\), and (iii) the non-extensive distributed concentration of electrons when \(\alpha=0\), respectively. Thus, the following normalized hydrodynamic continuity and momentum equations are obtained by implementing the mass and momentum conservation low along with the above plasma assumptions for positive and negative ions:  
\[\partial_{t}(N_{pi})+\partial_{z}(N_{pi}U_{pi})=0, \tag{2}\]\[\partial_{t}(N_{ni})+\partial_{z}(N_{ni}U_{ni})=0, \tag{3}\]  
\[\partial_{t}(U_{pi})+U_{pi}\partial_{z}(U_{pi})+\partial_{z}\Phi+\mu_{pi} \partial_{zz}(U_{pi})=0, \tag{4}\]  
\[\partial_{t}(U_{ni})+U_{ni}\partial_{z}(U_{ni})-M_{r}\partial_{z}\Phi+\frac{ \delta_{ei}}{N_{ni}}\partial_{z}(N_{ni})+\mu_{ni}\partial_{zz}(U_{ni})=0. \tag{5}\]  
The aforementioned plasma is bounded through the following normalized equation, which is obtained by considering the Maxwell equations along with the normalized electric field \(E=-\
abla\Phi\),  
\[\partial_{zz}\Phi=-N_{r2}\left\{[1+(q-1)\Phi]^{\frac{(q+1)}{2(q-1)}}\times \left[1+B_{1}\Phi+B_{2}\Phi^{2}\right]\right\}+N_{pi}-N_{ni}, \tag{6}\]  
where  
\[M_{r}=\frac{m_{pi}}{m_{ni}},\delta_{ei}=\frac{T_{e}}{\left(1-N_{r1}\right)T_{ ni}}. \tag{7}\]  
Here, \(N_{pi}\) (\(N_{ni}\)) and \(N_{e}\) are the densities of the positive (negative) ions and electrons normalized by \(N_{pi0}\), respectively. \(U_{pi}\) (\(U_{ni}\)) is the fluid positive (negative) ions velocity normalized by the positive ion acoustic speed \(C_{ips}=\sqrt{k_{B}T_{e}/m_{pi}}\sqrt{\left(N_{r1}+N_{r2}\right)/N_{r2}(1-N_{r 1})}\) based on the neutrality condition. \(\Phi\) is the normalized electrostatic potential by introducing \(\Phi\to e\phi/k_{B}T_{e}\). \(z\) and \(t\) are normalized by electron Debye length (\(\lambda_{De}=\sqrt{k_{B}T_{e}/4\pi N_{0}e^{2}}\)) and positive ion plasma frequency \(\omega_{pi}^{-1}=\lambda_{De}/C_{ips}\), respectively. \(\mu_{pi}\) (\(\mu_{ni}\)) is the normalized viscosity coefficient of positive (negative) ions that normalized by introducing \(\omega_{pi}^{-1}/m_{pi}N_{pi}C_{ips}^{2}\) (\(\omega_{pi}^{-1}/m_{ni}N_{ni}C_{ips}^{2}\)). Without loss of generality, it is assumed that \(T_{pi}\approx 0\) because \(T_{e}\) is modified by \(T_{e}/\left(1-N_{r1}\right)\)[4]. In addition, the ion acoustic mode is generated by the positive ion mode because the mass number (146) of \(SF_{6}^{-}\) is greater than the mass number (39) of (\(K^{+}\)).