## 4 Results and Discussion
(iv) Figures 5 and 6 show the influence of \(\alpha\) and \(q\), respectively, on the nonlinear IA single-, and overtaking collision of double-, triple-, as well as four-shock wave excitations by choosing other parameters constant. It is clearly shown form these figures that the amplitude and thickness of shock wave excitations decrease (increases) with increasing in \(q\) and \(\alpha\). It is also found that the compressive and rarefactive electrostatic of not only single- but also overtaking collision of multi-shock wave excitations are supported in the considered plasmas. (v) Figure 7 shows the rarefactive electrostatic single- and overtaking collision of multi-shock wave and behavior of its corresponding normalized electric field by taking \(N=1,2,3,4\) and different values of \(\alpha\) with \(q=1\), \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(N_{r1}=0.9\), \(N_{r2}=0.5\), \(\mu_{ni}=0.05\), and \(\mu_{pi}=0.35\), whereas Figure 8 shows the rarefactive electrostatic overtaking collision of multi-shocks and behavior of its corresponding normalized electric field by taking \(N=2,3\), different values of \(T\) and isothermal electrons, i.e., \(\alpha=0\), \(q=1\) with \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(N_{r1}=0.9\), \(N_{r2}=0.5\), \(\mu_{ni}=0.05\), and \(\mu_{pi}=0.35\). It is evidently found from these figures that the aforementioned plasma supports the rarefactive shock wave phenomena with the presence of isothermal electrons (\(\alpha=0\), \(q=1\)), which is in good agreement with the experimental investigations [4]. The rarefactive shock wave phenomena also exists with the presence of only nonthermal populations electrons (\(q=1\)). It is also observed that the electric field becomes not only single- but also overtaking collision of multi-hump-shaped by depending on the value of \(N\) in the shock wave solution and pulseable with increases in time, as is expected. Figure 3: Effect of \(\mu_{ni}\) on the electrostatic (**a**) single-, (**b**) overtaking collision of double-, (**c**) overtaking collision of triple-, and (**d**) overtaking collision of four-shock wave excitations, respectively, by choosing \(\alpha=0.3\), \(q=0.35\), \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(N_{r1}=0.5\), \(N_{r2}=0.02\), \(\mu_{pi}=0.5\), and \(T=5\). Figure 4: Effect of \(N_{r2}\) when \(N_{r1}=0.1\) on the electrostatic overtaking collision of (**a**) double-, and (**b**) triple-, and \(N_{r1}\) when \(N_{r2}=0.8\) on the electrostatic overtaking collision of (**c**) double-, and (**d**) triple-shock wave excitations, respectively, by choosing \(\alpha=0.3\), \(q=0.35\), \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(t_{pi}=0.35\), \(\mu_{ni}=0.05\), and \(T=10\). Figure 5: _Cont._  
Figure 5: Effect of \(q\) on the electrostatic (**a**) single-, (**b**) overtaking collision of double-, (**c**) overtaking collision of triple-, and (**d**) overtaking collision of four-shock wave excitations, respectively, by choosing \(\alpha=0.3\), \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(N_{r1}=0.5\), \(N_{r2}=0.02\), \(\mu_{ni}=0.01\), and \(\mu_{pi}=0.5\). Figure 6: Effect of \(\alpha\) on the electrostatic (**a**) single-, (**b**) overtaking collision of double-, (**c**) overtaking collision of triple-, and (**d**) overtaking collision of four-shock wave excitations, respectively, by choosing \(p=0.45\), \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(N_{r1}=0.5\), \(N_{r2}=0.02\), \(\mu_{ni}=0.01\), and \(\mu_{pi}=0.5\). Figure 8: _Cont._  
Figure 7: Rarefactive overtaking collision of electrostatic shock wave (lower curves) and its corresponding electric field behavior (upper curves) for (**a**) \(N=1\), (**b**) \(N=2\), (**c**) \(N=3\), and (**d**) \(N=4\) with \(\alpha=0\) (red color), 0.1 (blue color), 0.2 (green color), 0.3 (cyan color), \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(N_{r1}=0.9\), \(N_{r2}=0.5\), \(\mu_{ni}=0.05\), and \(\mu_{pi}=0.35\). On the other hand, the considered plasma system coincides with the plasma system that is proposed in [30] when \(\alpha=0\) and \(q=1\).