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Weakly nonlinear surface waves on the plasma-vacuum interface

We consider the free boundary problem for a plasma--vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement, where the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Our aim is to construct highly oscillating surface wave solutions in weakly nonlinear regime to this plasma--vacuum interface problem. 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.