Exact solutions to magnetogasdynamics using Lie point symmetries

In the present work, we find some exact solutions to the first order quasilinear hyperbolic system of partial differential equations (PDEs), governing the one dimensional unsteady flow of inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system whose simple solutions provide nontrivial solutions of the original system. Using this exact solution, we discuss the evolutionary behavior of weak discontinuities.