## 4 Results and Discussion  
To report the influences of \(N_{r1}\) (density ratio of negative to positive ions), \(N_{r2}\) (density ratio of electrons to positive ions), \(\mu_{pi}\) ( viscosity coefficient of positive ions), \(\mu_{ni}\) (viscosity coefficient of negative ions), \(\alpha\) (population of nonthermal electrons), and \(q\) (strength of non-extensive electrons) on the basic features of single and overtaking collision among multi-shock waves formation, an unmagnetized collosionless three-component plasma having positive as well as negative ions and \((\alpha,q)\)-distributed electrons is considered. The Burgers equation with the exact \(N\)-shocks solution as mentioned in Equation (21) is obtained to reveal such physical phenomena. In the presented analysis, the physical parameters are taken as \(M_{r}=3.74\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(0<N_{r1}<1\), \(0<N_{r2}<1\), \(\mu_{pi}=0.1\sim 0.5\), and \(\mu_{ni}=0.001\sim 0.4\) based on the work in [4,30] and \(-1<q<1\) (for superthermality), \(q>1\) (subthermality), \(q=1\) (isothermality), and \(0<\alpha<1\) based on the work in [25]. (i) Figure 1a,b shows, respectively, the variation of phase velocity with regards to \(N_{r2}\) and \(\alpha\) for different values of \(N_{r1}\) and \(q\). It is observed from Figure 1 that the linear phase velocity (\(V_{p}\)) losses (gains) energy with increases the density ratio of electrons to positive ions (the density ratio of negative to positive ions). Whereas, \(V_{p}\) loses (gains) energy with increasing \(q\) for the case of subthermality (superthermality). Moreover, \(V_{p}\) gains more energy with increases the population of nonthermal nonextensive electrons. (ii) Figure 2b-d demonstrates the influence of \(\mu_{pi}\) on the nonlinear IA single-, and overtaking collision of triple-, as well as five-shock wave excitations, while Figure 2a demonstrates the IA single-shock wave excitation for different values of time \(T\) by choosing the other parameters constant. Besides, the influence of \(\mu_{ni}\) on the nonlinear IA single-, and overtaking collision of double-, triple-, as well as four-shock wave excitations by choosing the other parameters constant is demonstrated in Figure 3. It is evidently reveled from these figures that both \(\mu_{pi}\) and \(\mu_{ni}\) play a vital role in the formation of not only nonlinear electrostatic single-, but also overtaking collision of multi-shock waves in the negative ions plasmas. Because, the amplitude and thickness of monotonically shock wave remarkably increases with increases in \(\mu_{pi}\), but very sightly increases with increases in \(\mu_{ni}\). Besides, the amplitude and thickness of shock wave remains unchanged, but the position of shock wave changes with increases in time. Figure 3: _Cont._  
Figure 2: **(a)** Electrostatic single-shock wave excitation due to the variation of \(T\) with \(\mu_{pi}=0.3\) and effect of \(\mu_{pi}\) on the electrostatic (**b**) single-, (**c**) overtaking collision of triple-, and (**d**) overtaking collision of five-shock wave excitations with \(T=5\), respectively. The other parameters are chosen as \(\alpha=0.3\), \(q=0.35\), \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(N_{r1}=0.5\), \(N_{r2}=0.02\), and \(\mu_{ni}=0.01\). (iii) Figure 4 displays the influence of \(N_{r2}\) (Figure 4a,b) and \(N_{r1}\) (Figure 4c,d) on the nonlinear electrostatic overtaking collision of double- and triple-shock wave excitations by assuming other parameters are constant. It is found from Figure 4 that both density ratios remarkably play distinct roles to each other in the formation of overtaking collision of \(N\)-shocks in the considered plasmas. This is because the monotonically shock wave also occurred, in which the amplitude and thickness of shocks decreases monotonically with increases in \(N_{r2}\). However, the amplitude and thickness of shocks increases with increases in \(N_{r1}\).