  We compute electromagnetic wave propagation through the magnetosphere of a
magnetar. The magnetosphere is modeled as the QED vacuum and a cold, strongly
magnetized plasma. The background field and electromagnetic waves are treated
nonperturbatively and can be arbitrarily strong. This technique is particularly
useful for examining non-linear effects in propagating waves. Waves travelling
through such a medium typically form shocks; on the other hand we focus on the
possible existence of waves that travel without evolving. Therefore, in order
to examine the nonlinear effects, we make a travelling wave ansatz and
numerically explore the resulting wave equations. We discover a class of
solutions in a homogeneous plasma which are stabilized against forming shocks
by exciting nonorthogonal components which exhibit strong nonlinear behaviour.
These waves may be an important part of the energy transmission processes near
pulsars and magnetars.
