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 Alfvén 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 ) ) .
The propagation of a single Alfvén mode is unaffected but QED does alter the coupling between the Alfvén modes .

This processes may have important consequences for the study of neutron-star magnetospheres especially if the typical magnetic field strength exceeds the QED critical value ( B _ { \hbox { \scriptsize QED } } \approx 4.4 \times 10 ^ { 13 } G ) as is suspected for soft-gamma repeaters and anomalous X-ray pulsars .