## 1 Introduction  
Dust-acoustic waves (DAWs) have many industrial applications such as laboratory plasma devices, semiconductor chips, and fusion reactors [1; 2; 3; 4]. They are also commonly observed in various space environments such as asteroid zones, the lower and upper mesosphere and ionosphere of the Earth, cometary tails, interplanetary spaces, and interstellar media [5; 6; 7; 8; 9]. The numerous applications of DAWs have motivated researchers to study these waves for different plasma conditions. For instance, [PERSON] et al. [10] studied dusty ion acoustic waves (DIAW), and their results were compared with observations of [PERSON]'s Comet. [PERSON] et al. [11] showed, theoretically, that the propagation of low energetic acoustic waves in both linear and nonlinear modes is due to the existence of dusty particles. [PERSON] et al. [12; 13] studied low energetic acoustic waves under microgravity conditions. Looking for a different way of propagating acoustic waves in dusty plasma was continued by many researchers [14; 15; 16; 17; 18; 19]. The propagation of nonlinear dust-acoustic (DA) solitary waves in unmagnetized dusty plasma composed of Maxwellian electrons, vortex-like (trapped) ions, and cold mobile dust grain was studied by [PERSON] et al. [20]. The Korteweg-de Vries (KdV) equation, which emits the solitary wave solution and was derived using the standard reductive perturbation method for positively and negatively charged warm dust species and ions, was found to have nonthermal distribution by [PERSON] and [PERSON] [21]. The reductive perturbation method, used to study shock waves that may exist in a dusty plasma system, is composed of four components via the derivation of Burgers' equation under the effect of viscosity, which causes the dissipation of energy in the system under consideration [22]. The possibility of observing shock waves related to the dust-charging process was studied for three components of dusty plasma [23]. The reductive perturbation method was used to derive the KdV-Burgers' equation to study the effects of dust size distribution on the propagation of shock waves that propagate in a dusty plasma system composed of negatively dust species, two types of ions that differ in temperature, and electrons [24]. [PERSON] et al. studied the propagation of dust-acoustic solitary waves and double-layer waves for four-component dusty plasma system containing nonextensive distributed electrons and ions via Sagdeev pseudopotential and using the traditional reductive perturbation method to derive KdV and modified KdV equations [25; 26; 27]. Suprathermal populations are ubiquitous in space plasmas. The observed particle velocity distributions from space show superthermal, high-energy tails, which take the formula as a power law. As such, the so-called kappa distributions fit very well into the observed populations of plasma species [28]. In the mid-1960s, the formulas of kappa distributions were firstly introduced by [PERSON], [PERSON], and [PERSON] [29; 30; 31]. The effects of kappa distribution parameters and viscosity on the polarity of shock waves propagated in four-component dusty plasma composed of positive and negative cold dust species and kappa distributed electrons and ions were studied by [PERSON] and [PERSON] [32]. In this paper, the effects of both thermal motion and viscosity on the inertial species in a four-component, collision-less, unmagnetized dusty plasma system were investigated, while the effects of kappa parameters for inertialess species were introduced. Thus, a plasma system under consideration contains negative and positive dust species as well as kappa-distributed electrons and ions, and this system can be found in both space environments and laboratories [33; 34; 35]. This paper is arranged as follows: Section 2 presents the governing equations that account for the four-component dusty plasma system. The momentum equation contains the effect of the viscosity on both the positive and negative dust species.