In this talk we consider the free boundary problem in three space dimensions 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 weakly nonlinear, highly oscillating solutions to this plasma-vacuum interface problem. Under a necessary and sufficient stability condition for a piecewise constant background state, we construct by a geometric optics approach approximate solutions, at any arbitrarily large order of accuracy, when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the plasma-vacuum interface solution give rise to surface waves on either side of the interface. Such waves decay exponentially in the normal direction to the interface and, in the weakly nonlinear regime that we consider here, their leading amplitude is governed by a nonlocal Hamilton-Jacobi type equation, as for Rayleigh waves in elastodynamics and current-vortex sheets in MHD.
This is a joint work with Yuan Yuan (CAMIS, South China Normal Univ.).