The bifurcation analysis of the MHD Rankine–Hugoniot equations for a perfect gas

Abstract

This article provides a complete bifurcation analysis of the Rankine–Hugoniot equations for compressible magnetohydrodynamics (MHD) in the case of a perfect gas. Particular scaling properties of the perfect-gas equation of state are used to reduce the number of bifurcation parameters. The smaller number, together with a novel choice, of these parameters results in a detailed picture of the global situation which is distinctly sharper than the one implied by previous literature. The description includes statements about the location, topology, and dimensions of various regimes corresponding to different combinations of possible shock waves of given type, in dependence of the adiabatic exponent of the gas. The analysis is also a prerequisite for new results on the existence and bifurcation of viscous profiles for intermediate MHD shock waves that are presented in a separate paper.