  We examine wave propagation and the formation of shocks in strongly
magnetized plasmas by applying a variational technique and the method of
characteristics to the coupled magnetohydrodynamic (MHD) and
quantum-electrodynamic (QED) equations of motion. In sufficiently strong
magnetic fields such as those found near neutron stars, not only is the plasma
extremely relativistic but the effects of QED must be included to understand
processes in the magnetosphere. As Thompson & Blaes [1] find, the fundamental
modes in the extreme relativistic limit of MHD coupled with QED are two
oppositely directed Alfv\'{e}n modes and the fast mode. QED introduces
nonlinear couplings which affect the propagation of the fast mode such that
waves traveling in the fast mode evolve as vacuum electromagnetic ones do in
the presence of an external magnetic field [2] (Heyl & Hernquist 1998). The
propagation of a single Alfv\'{e}n mode is unaffected but QED does alter the
coupling between the Alfv\'{e}n modes.
