Geometric Optics for Surface Waves on the Plasma–Vacuum Interface: Higher Order Expansion

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 necessary 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 nontrivial residual non-oscillatory components. The content of this chapter summarizes the result in Secchi and Yuan (Secchi P, Yuan Y (2022) Weakly nonlinear surface waves on the plasma–vacuum interface. J Math Pures Appl 163:132–203).