  Hall-MHD is a mixed hyperbolic-parabolic partial differential equation that
describes the dynamics of an ideal two fluid plasma with massless electrons. We
study the only shock wave family that exists in this system (the other
discontinuities being contact discontinuities and not shocks). We study planar
travelling wave solutions and we find solutions with discontinuities in the
hydrodynamic variables, which arise due to the presence of real characteristics
in Hall-MHD. We introduce a small viscosity into the equations and use the
method of matched asymptotic expansions to show that solutions with a
discontinuity satisfying the Rankine-Hugoniot conditions and also an entropy
condition have continuous shock structures. The lowest order inner equations
reduce to the compressible Navier-Stokes equations, plus an equation which
implies the constancy of the magnetic field inside the shock structure. We are
able to show that the current is discontinuous across the shock, even as the
magnetic field is continuous, and that the lowest order outer equations, which
are the equations for travelling waves in inviscid Hall-MHD, are exactly
integrable. We show that the inner and outer solutions match, which allows us
to construct a family of uniformly valid continuous composite solutions that
become discontinuous when the diffusivity vanishes.
