Exact solutions to the ideal magneto-gas-dynamics equations through Lie group analysis and substitution principles.

In this paper, we consider the equations governing an inviscid, thermally non-conducting fluid of infinite electrical conductivity in the presence of a magnetic field and subject to no extraneous force. By using conveniently the Lie point symmetries admitted, we map the governing system into an equivalent autonomous form. The transformed system can be directly inspected in order to find some simple solutions that, written in terms of the original variables, provide non-trivial exact solutions of the system at hand. Use is made of some finite transformations known in the literature as substitution principles, enabling us to build exact solutions containing some arbitrary functions. The link between the substitution principle and the recently discovered Bogoyavlenskij symmetries for equilibrium magnetohydrodynamics is also discussed. Some of the recovered solutions are considered to solve well-known physically relevant boundary value problems and the linear stability analysis is performed, thus generalizing well-established results.