A Study on the Exact Solutions of the Ramani Equation Using Lie Symmetry Analysis

Shock waves propagating in a plasma medium are modelled by the sixth-order nonlinear Ramani equation. All possible closed-form solutions to the Ramani equation are derived through the holistic approach using the Lie symmetry method. This involves computing Lie point symmetries and the corresponding similarity reductions. However, certain symmetry-reduced ordinary differential equations possess no point symmetries. We investigate such equations through singularity analysis. Their solutions are obtained as right Painlevé series. In addition, we present the travelling-wave solutions of the original equation via the improved \(\left( \frac{G^\prime }{G}\right) \) $\left(\frac{G^{\prime }}{G}\right)$ -expansion method. The soliton-type, periodic, and hyperbolic graphs of the solutions which aid in comprehending the wave phenomenon is displayed.