Collision of shock waves in plasma medium: a travelling-wave solution

The collision of a pair of shock waves in a plasma medium is described by the coupled Ramani equation, \(\begin{aligned}&\qquad \qquad \qquad \qquad u_{xxxxxx}+15u_{xx}u_{xxx}+15u_{x}u_{xxxx}+45{u_x}^2u_{xx}-5u_{xxxt}-15u_{xx}u_t -15u_xu_{xt}+25u_{tt}+18w_x = 0, \nonumber \\&\qquad \qquad \qquad \qquad w_t - w_{xxx} - 3w_xu_x - 3wu_{xx} = 0. \end{aligned}\) $$\begin{array}{cc}& u_{\mathit{xxxxxx}}+15u_{\mathit{xx}}{u}_{\mathit{xxx}}+15u_x{u}_{\mathit{xxxx}}+45{u_x}^2{u}_{\mathit{xx}}-5u_{\mathit{xxxt}}-15u_{\mathit{xx}}{u}_t-15u_x{u}_{\mathit{xt}}+25u_{\mathit{tt}}+18w_x=0,\hfill \\ \hfill & w_t-w_{\mathit{xxx}}-3w_x{u}_x-3w{u}_{\mathit{xx}}=0.\hfill \end{array}$$ This equation is a nonlinear evolution equation, and its travelling-wave solution in the exact form provides insight into its various physical phenomena. When evaluating the accuracy of more advanced versions of these equations when solved numerically, these precise analytical answers serve as a basis for further investigation. The improved \(({G'}/{G})\) $(G^{\prime }/G)$ -expansion approach is used to discuss travelling-wave solutions of the coupled Ramani problem in this study. In terms of rational, trigonometric and hyperboloic functions, exact solutions are found. Studying the physically important soliton solutions of the coupled Ramani problem is made possible by this analytical treatment of the equation.