In this article, we study a $(2 + 1)$-dimensional non-linear Vakhnenko equation which appears in a relaxing high rate active barothropic medium. The Lie symmetry analysis is applied to study the solution of $(2+1)$-dimensional Vakhnenko equation and also used to find the optimal system of one-dimensional Lie subalgebra. The two-dimensional non-linear Vakhnenko equation is reduced to various ordinary differential equations by applying similarity transformation method, and solving these reduced ordinary differential equations, we find invariant solutions of $(2 + 1)$-dimensional Vakhnenko equation. Moreover, the adjoint equation of the Vakhnenko equation and its conservation laws are also obtained, which can be used for mathematical analysis.

The singular surface theory is used for the planar and cylindrically symmetric flow of a dusty gas in radiative magnetogasdynamics to study the behaviour of different modes of wave propagation, and their culmination into a steepened form. We have considered one-dimensional steepening of waves. The transport equation for the jump discontinuity in the velocity gradient at the wave heads is determined and solved numerically. The effects of the van der Waals excluded volume of the medium, the mass concentration of solid particles, the ratio of densities, the ratio of specific heats, the radiation parameter and the magnetic field strength on the shock formation are analysed in detail. It is observed that the gradient of density, velocity, pressure and magnetic field are related to each other.