On the Structure of MHD Shock Waves in Diffusive-Dispersive Media

We consider the equations of compressible magnetohydrodynamics including the diffusive effects of fluid viscosity, thermal heat conductivity, electrical resistivity, and, in particular, the dispersive influence of the Hall term. The equations describe the dynamics of a plasma as it arises in astrophysical and technical applications. For a model problem we prove an existence result for shock profiles for all values of the Hall parameter. The same question for the complete six-dimensional system seems to be only solvable for small values of the Hall parameter, that is, viewing the Hall effect as a perturbation. To obtain substantial information on the complete system for all ranges of the dissipation parameters we present a careful numerical study. In particular the study confirms a well-known conjecture on the existence and bifurcation of orbits and illustrates the breaking of symmetry due to the Hall term.