## 3 Mathematical Formulation
### N-Shock Wave Solutions  
Before determining the useful \(N\)-shock wave solution of Equation (18), one first needs to transform Equation (18) to the well-established standard type Burgers equation by using suitable variables transformation. To do so, one can easily convert Equation (18) to the following standard type Burgers equation by considering the transforms \(X=Y(C/A)^{1/2}\) and \(\Phi=U(B/A)^{-1}(C/A)^{1/2}\) into account,  
\[\frac{\partial U}{\partial T}+U\frac{\partial U}{\partial Y}+\frac{\partial^ {2}U}{\partial Y^{2}}=0, \tag{20}\]  
where the nonlinear and dissipative coefficients are unity. Equation (20) is also converted to a heat type equation by linearizing with the well-known Cole-Hopf transformation [42] as \(U\to 2\frac{\partial}{\partial Y}(\ln U(Y,T))\). The heat type equation provides a simple solution as \(V_{i}(Y,T)=\exp(k_{i}Y-\omega_{i}T)\), where \(i=1,2,3,\ldots N\), \(k_{i}\) is the wave number, and \(\omega_{i}\) is the frequency and their linear superposition as \(V(Y,T)=K+\sum_{i}^{N}[\exp(k_{i}Y-\omega_{i}T)]\) (K is real constant) is also solution. In addition, one can easily derive the multi-shocks of Equation (20) by considering \(U=(\frac{\partial V}{\partial Y}/V)\). Based on the above assumptions, a few authors [42; 43; 44] have proposed the multi-shocks solutions of only the standard Burgers equation having nonlinear and dissipative coefficients unity, like \(U_{T}+UU_{x}=U_{xx}\). Besides, the nonlinear and dissipative coefficients of the Burgers equation are not in unity, but strongly dependent on the physical parameters. Therefore, we still require the \(N\)-shock wave solutions of Equation (18) to understand the nature of multi-shocks in the aforementioned plasmas with the influences of plasma parameters. It is noted that Equation (18) can also be reduced to a heat type equation by introducing \(\Phi(X,T)\to 2((B/A)^{-1}(C/A)^{1/2}\frac{\partial}{\partial X}(\ln(\Phi((C/A)^{-1/2 }X,T))\). Besides, one can easily derive the dissipation relation of Equation (18) as \(\omega_{i}=k_{i}^{2}\) by inserting \((\exp(k_{i}X-\omega_{i}T))\), \((i=1,2,\ldots N)\) with the above transform directly in Equation (18). Therefore, the useful \(N\)-shock wave solution of Equation (18) is defined as  
\[\Phi(X,T)=\frac{A^{3/2}C^{1/2}}{B}\left\{\frac{2\sum_{i=1}^{N}\left[k_{i}\exp \left(\frac{k_{i}X}{\sqrt{(C/A)}}-k_{i}^{2}T\right)\right]}{1+\sum_{i=1}^{N} \left[\exp\left(\frac{k_{i}X}{\sqrt{C/A}}-k_{i}^{2}T\right)\right]}\right\}, \tag{21}\]  
where \(N\) is a positive integer number, \(A=\frac{2}{V_{p}^{3}}+\frac{2N_{r1}V_{p}M_{r}}{\left(V_{p}^{2}-\delta_{ei} \right)^{2}}\), \(B=\frac{3}{V_{p}^{2}}-\frac{3N_{r1}V_{p}^{2}M_{r}^{2}}{\left(V_{p}^{2}-\delta_ {ei}\right)^{3}}+\frac{\delta_{ii}N_{r1}M_{r}^{2}}{\left(V_{p}^{2}-\delta_{ei} \right)^{3}}-2N_{r2}K_{2}\), and \(C=\frac{\mu_{pi}}{V_{p}^{3}}+\frac{\mu_{ni}V_{p}M_{r}}{\left(V_{p}^{2}-\delta _{ei}\right)^{2}}\). It is obviously found from Equation (21) that the electrostatic \(N\)-shocks are only dependent on the plasma parameters, which indicates that Equation (21) is very useful for studying the nature of \(N\)-shocks in plasmas. Equation (21) is also checked by inserting in Equation (18)with the aid of computational software, like Maple 18, and found correct. One can easily study the nature of electrostatic shocks and their corresponding electric filed (\(E=-\
abla\Phi\)) by using the analytical stationary solution as in Equation (21) of the Burgers-like Equation, as in Equation (18).