## 4 Results and Discussion
The appearance of both positive and negative shock waves, which are provided by the graph of the electrostatic potential, \(\phi_{B}(\xi,\tau)\), via the space parameter, \(\xi\), for different values of the kappa parameter, \(k\), and the other parameters taken from Figures 1 and 2 are shown in Figure 3. The effects of viscosity for both positive and negative dust components are shown to be able to plot the positive electrostatic potential, \(\phi_{B}(\xi,\tau)\) (at \(k=2.5,\ \text{and}\ \sigma=0\)) in Figure 4. The graphs in Figure 4 show that the shapes of the positive shock waves are changed due to varying the viscosity  
Figure 1: The 2D soliton wave electrostatic potential yielded by the KdV equation at different values of kappa distribution parameters \(k\) and with \(\sigma=0\). Figure 2: The 2D soliton wave electrostatic potential yielded by the KdV equation for different values of the thermal parameter \(\sigma\). Figure 5 shows that positive and negative solitary waves are propagated by the interaction between a nonlinear term with dispersion and dissipation, taking the introduced arbitrary parameters as \(\alpha=\frac{1}{2}\), \(\gamma=\frac{1}{2}\). In this case, Equation (9) has a solitary wave solution:  
\[ \begin{split}&\phi_{B}(\xi,\tau)=\phi_{\text{KdV}-B}\Big{(}1-e^{C_{1 }(\xi-\tau\tau)}\cos(C_{2}(\xi-\
u\tau))\Big{)},\\ &\phi_{0\text{KdV}-B}=2\
u/A,\,C_{1}=-C/2B,\,\,\,\text{and}\,\,\,C _{2}=\sqrt{\frac{\
u}{B}}.\end{split}  \tag{18}\]  
From the solutions of the KdV Equation (10f), Burgers' Equation (10g), and the KdV-Burgers Equation (18), we can say that the types (compressive or rarefactive) of propagated waves are controlled by the values of nonlinear term coefficient \(A\). Compressive waves (soliton, shock, and solitary) are propagated at \(A>0\), while rarefactive waves are propagated at \(A<0\). At \(A=0\), the propagated wave cannot be defined. Thus, the strength of the nonlinear term must be increased. The so-called modified KdV-Burgers and Further-Burgers equations are derived at \(\beta=2\), with the dispersion and dissipation coefficient depending mainly on the arbitrary parameters \(\alpha\) and \(\gamma\). Finally, as shown from the introduced stretching and derived equations, there are three parameters that can directly affect the strength of nonlinearity and the appearance of the effects of both dispersion and dissipation in the medium, in addition to the propagation of compressive and rarefactive waves, which depend mainly on the kappa parameter values, as shown in Figures 1-5. As it appears from the form of the dissipation coefficient, the dissipation of energy depends mainly on the dust fluid viscosity. This analysis may be useful in predicting the types of propagated dust-acoustic waves in the auroral zone of the ionosphere, for laser-produced plasma, and for interstellar dust plasma where non-Maxwellian distributions are considered. The chosen plasma parameters are relevant to different areas of space science, such as Jupiter's magnetosphere, Earth's mesosphere, and cometary tails [33; 34; 35]. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable, this research not involving humans or animals. **Informed Consent Statement:** Not applicable. Figure 5: The 2D oscillatory wave electrostatic potential yielded by the solution of the KdV–Burgers’ equation for different values of \(k\), with \(\eta_{1}=0.6,\sigma=0,\eta_{2}=0.7\).