## 2 Governing Equations  
In our multicomponent complex plasma system, the negative and positive dust species are taken as two fluids, while the light species, i.e., electrons and ions, are assumed to be kappa distributed. The following normalized continuity and momentum equations were used to study the motion of the two fluids [25; 26] after adding the effect of viscosity and taking the distribution of inertialess species via kappa distribution:  
\[\frac{\partial}{\partial t}N_{1}+\frac{\partial}{\partial x}[N_{1}V_{1}]=0, \tag{1a}\] \[\frac{\partial}{\partial t}V_{1}+V_{1}\frac{\partial}{\partial x}V_{1}-\frac{ \partial}{\partial x}\phi+\eta_{1}\frac{\partial^{2}}{\partial x^{2}}V_{1}=0,\] (1b) \[\frac{\partial}{\partial t}N_{2}+\frac{\partial}{\partial x}[N_{2}V_{2}]=0,\] (2a) \[\frac{\partial}{\partial t}P+V_{2}\frac{\partial}{\partial x}P+3P\frac{ \partial}{\partial x}V_{2}=0,\] (2b) \[\mu\bigg{[}\frac{\partial}{\partial t}V_{2}+V_{2}\frac{\partial}{\partial x}V_{2 }\bigg{]}+\frac{\partial}{\partial x}\phi+\frac{\sigma}{N_{2}}\frac{\partial} {\partial x}P+\eta_{2}\frac{\partial^{2}}{\partial x^{2}}V_{2}=0. \tag{2c}\]  
In order to make a self-consistent system, the following [PERSON]'s equation is used:  
\[\frac{\partial^{2}}{\partial x^{2}}\phi-N_{1}+\mu_{+}N_{2}+\mu_{i}N_{i}-\mu_{ e}N_{e}=0 \tag{3}\]  
where \(N_{i}(x,t)\) and \(N_{e}(x,t)\) are the number densities of ions and electrons, respectively. In this paper, the normalized densities of ions and electrons are provided in kappa distribution forms as  
\[N_{i}=\bigg{(}1+\frac{\phi}{k_{i}-3/2}\bigg{)}^{-(k_{i}-1/2)}, \tag{4a}\]\[N_{e}=\left(1-\sigma_{e}\frac{\phi}{k_{e}-3/2}\right)^{-(k_{e}-1/2)} \tag{4b}\]  
In Equations (1)-(3), \(N_{1}(x,t)(N_{2}(x,t))\) and \(V_{1}(x,t)(V_{2}(x,t))\) are, respectively, the negative (positive) normalized densities by their equilibrium values, \(N_{10}(N_{20})\), and the negative (positive) normalized velocity by the dust-acoustic speed, \(C_{d}=\sqrt{\mu}v_{T}\); \(\mu=Z_{1}m_{2}/(Z_{2}m_{1})\), \(v_{T}=(k_{B}T_{i}Z_{2}/m_{2})^{1/2}\) is the thermal velocity, \(m_{2}(m_{1})\) and \(Z_{2}(Z_{1})\) are the rest mass of positive (negative) dust fluid and the number of charges on the positive (negative) dust grain surface, \(T_{i}\) is the ion temperature, and \(k_{B}\) is the Boltzmann constant. \(P(x,t)\) and \(\phi(x,t)\) are the normalized pressure and electrostatic potential by \(N_{20}k_{B}T\) and \(\left(k_{B}T_{i}/e\right)\), respectively. Space and time are normalized by the dusty Debye length (\(\lambda_{D}=\left[k_{B}T_{i}/\left(4\pi e^{2}N_{10}Z_{1}\right)\right]^{1/2}\)) and the inverse of the dusty plasma frequency \(\omega_{-}^{-1}=\left[m_{1}/\left(4\pi N_{10}e^{2}Z_{1}^{2}\right)\right]^{1/2}\), respectively. \(\sigma_{e}=T_{i}/T_{e}\) and \(\sigma=T_{2}/\left(Z_{2}T_{i}\right)\) is the temperature ratio between the positive dust and ions, where \(T_{e}\) is the electron temperature. \(\mu_{+}=N_{20}Z_{2}/(N_{10}Z_{1})\), \(\mu_{e}=N_{\omega 0}/(N_{10}Z_{1})\), and \(\mu_{i}=N_{i0}/(N_{10}Z_{1})\), which satisfy the neutrality condition, and \(\mu_{+}-\mu_{e}+\mu_{i}=1\), \(N_{\omega 0}\), and \(N_{i0}\) are the densities of electrons and ions at equilibrium, respectively. \(\eta_{1}(\eta_{2})\) is the normalized viscosity of negative (positive) massive species. \(\kappa_{i}(\kappa_{e})\) is the kappa parameter of ions (electrons).  
Equations (1)-(4) are complex, nonlinear, partial differential equations of six unknown functions: \(N_{1}\), \(V_{1}\), \(N_{2}\), \(P\), \(V_{2}\), and \(\phi\). Therefore, in the next section, we perform a small amplitude wave analysis by using the reductive perturbation method (RPM) [36; 37].