## 5 Concluding Remarks  
A plasma system consisting of positive as well as negative ions and electrons is proposed to study the nature of not only single, but also overtaking collisions of multi-shocks electrostatic potential structures with the influences of plasma parameters, where electrons are assumed to follow not only isothermality but also superthermality, subthermality, nonthermality, and nonthermality with the presence of nonextensitivity. The \((\alpha,q)\)-velocity distribution function is considered because it is very effective in all cases of thermality conditions. By employing the well-established reductive perturbation approach, the Burger equation is obtained which divulges the shocks in the proposed plasmas. The new exact \(N\)-shock wave solution is archived by assuming directly the exponential function. It is found that \(\mu_{pi}\), \(\mu_{ni}\), and \(N_{\text{r2}}\) are significantly affected on the formation of monotonically single, and overtaking collisions of \(N\)-shocks by playing different roles. This is because the steepness and amplitude of shocks are sensitively increased (decreased) with the increase of \(\mu_{pi}\) (\(N_{\text{r2}}\)), but very slightly increased with the increase of \(\mu_{ni}\). In addition, both compressive and rarefactive single as well as overtaking collisions of \(N\)-shocks exist with the influences of parameters. The density ratios and strength of non-extensive as well as the population of nonthermal electrons are also significantly contributed in the formation of overtaking collisions of \(N\)-shocks in the plasmas. Besides, the electric field is produced overtaking collisions of \(N\)-hump shape structures due to the overtaking collisions of \(N\)-shocks electrostatic potential. Thus, the outcomes presented in this work would be very useful to understand the dynamics of overtaking collisions of electrostatic \(N\)-shocks and their corresponding electric field behaviors  
Figure 8: Rarefactive electrostatic overtaking collision of shock wave (lower curves) and its corresponding electric field behaviors (upper curves) for (**a**) \(N=2\) and (**b**) \(N=3\) with \(T=10\) (red color), \(15\) (blue color), \(20\) (green color), \(25\) (cyan color), \(\alpha=0\) and \(q=1\), \(M_{r}=3.75\), \(T_{ni}=0.05\), \(T_{e}=0.2\), \(N_{\text{r1}}=0.9\), \(N_{\text{r2}}=0.5\), \(\mu_{ni}=0.05\), and \(\mu_{pi}=0.35\). in many space and astrophysical environments [14; 15; 16; 17; 29] (especially, the electrostatic overtaking collisions of multi-shock noise in the Earth's ionosphere) and future experimental studies [4; 11; 12]. It is noted that the basic features of small but finite amplitude single- and overtaking collisions of multi-shocks are validated only if the nonlinear coefficient \(B\
eq 0\) is used, because the electrostatic potential function as mentioned in Equation (21) becomes infinite when \(B\to 0\). Further theoretical studies in the formation of overtaking collisions of \(N\)-shocks around the critical values by deriving a NLEE with higher order correction is still required. This will be our main focus for future studies. Conceptualization, M.G.H. ; methodology, P.A. and M.G.H. ; data curation, M.G.H. ; formal analysis, S.A.A.K. ; funding acquisition, S.A.A.K. ; investigation, M.G.H. and P.A. ; project administration, S.A.A.K. ; resources, M.G.H. ; software, M.G.H. ; supervision, M.G.H. and S.A.A.K. ; validation, P.A. ; visualization, M.G.H. and S.A.A.K. ; writing--original draft, S.A.; writing--review and editing, M.G.H. and S.A.A.K. All authors have read and agreed to the published version of the manuscript. This work is fully supported by Universiti Teknologi PETRONAS (UTP) and Ministry of Education, Malaysia through FRGS/1/2018/STG06/UTP/03/1/015 MA0-020 (New rational quartic spline for image refinement) and UTP through a research grant YUTP: 0153 AA-H24 (Spline Triangulation for Spatial Interpolation of Geophysical Data).