## 4 Results and Discussion  
In this paper, the reductive perturbation method in its generalized form (5) was used to derive some important nonlinear partial differential evolution equations (the KdV-Burgers family) to theoretically study some important waves, such as soliton, shock, and solitary waves, for unmagnetized, collisionless four-component dusty plasma, as taken from [25; 26], after adding the effect of viscosity and taking the distribution of inertialess particles as a kappa distribution. Some arbitrary parameters were presented in a reductive perturbation method to control the strength of nonlinearity, dispersion, and dissipation in the system under consideration. As shown in the theoretical analysis for the system under consideration, the existence of three arbitrary parameters is very important: \(\alpha\), which controls the strength of dispersion; \(\beta\), which controls the strength of nonlinearity; and \(\gamma\), which controls the strength of dissipation in our system. The nonlinear KdV-Burgers equation (9) can be obtained at \(\beta=1\), and this equation is a very important in plasma physics because it is a combination of two important equations, that is, the KdV and Burgers equations. This equation combines the nonlinear, dispersion, and dissipation terms in one equation with their coefficients depending on the arbitrary parameters \(\alpha\) and \(\gamma\). At \(\alpha=\frac{1}{2},\;\gamma>\frac{1}{2}\), i.e., the dissipation effect can be neglected in comparison with the dispersion, and the resultant equation is called the KdV evaluation equation, which balances between nonlinearity and disperses a soliton solution given by (10f), while at \(\alpha=1,\;\gamma=0\) (in this case, the effect of dissipation is larger than that of dispersion), in Equation (9) it can be converted to the so-called Burgers' equation, which emits a shock wave solution given in (10g). As shown from the soliton solution of the KdV Equation (10f) and the shock wave solution for Burgers' Equation (10g), the sign of the amplitudes of the propagated waves depends on the sign of the nonlinear coefficient, \(A\), as given in (10a), which depends mainly on the configurationally plasma parameters. The plotting of electrostatic potential, \(\phi_{KdV}(\xi,\tau)\), given by (10f) at \(\sigma=0.\sigma_{e}=0.5,\;\mu_{e}=0.2,\;\mu_{2}=0.5,\;\mu=2,\;\tau=0.01,\;v=0.04\) and at different values of the kappa parameter, \(k\), shows the propagation of negative (rarefactive) solitons at \(k<1.5\) and positive (compressible) solitons at \(k>1.5\) in Figure 1. The effects of the kappa parameter, \(k\), are obvious in Figure 1 in the shape (width and amplitude); that is, at \(k>1.5\), the change in both width and amplitude for the compressive soliton is very sensitive to the change in the kappa parameter, while the shape of the rarefactive soliton is slightly changed by the kappa parameter in comparison with that of the compressive soliton. The effects of the thermal motion of the positive dust fluid on the propagation of the compressive parameter at \(k=2.5\) and the rarefactive solitons at \(k=1\) are shown in Figure 2. It is clear from the graph that as the thermal parameter, \(\sigma\), increases, both the width and amplitude decrease, but the compressive soliton is affected by the thermal parameter, which is higher than that of the rarefactive parameter. At \(\alpha=1\), \(\gamma=0\), the effect of dispersion can be neglected in comparison with the dissipation effect in this case, and absent the effect of thermal motion, the system under consideration is converted to that of [32].