Geometric Optics for Surface Waves on the Plasma–Vacuum Interface

We consider the free boundary problem for a plasma–vacuum interface
in ideal incompressible magnetohydrodynamics, where the Maxwell equations for
electric and magnetic fields are considered in the vacuum region. Under a neces-
sary and sufficient stability condition for a piecewise constant background state, we
construct approximate solutions at any arbitrarily large order of accuracy to the free
boundary problem in three space dimensions when the initial discontinuity displays
high frequency oscillations. Moreover, such approximate surface waves have non-
trivial residual non-oscillatory components. The content of this paper summarizes
the result in Secchi and Yuan (2022).