  This work is devoted to the construction of weakly nonlinear, highly
oscillating, current vortex sheet solutions to the incompressible
magnetohydrodynamics equations. Current vortex sheets are piecewise smooth
solutions to the incompressible magnetohydrodynamics equations that satisfy
suitable jump conditions for the velocity and magnetic field on the (free)
discontinuity surface. In this work, we complete an earlier work by Ali and
Hunter and 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. As evidenced in
earlier works, high frequency oscillations of the current vortex sheet give
rise to `surface waves' on either side of the sheet. Such waves decay
exponentially in the normal direction to the current vortex sheet and, in the
weakly nonlinear regime that we consider here, their leading amplitude is
governed by a nonlocal Hamilton-Jacobitype equation known as the `HIZ equation'
(standing for Hamilton-Il'insky-Zabolotskaya) in the context of Rayleigh waves
in elastodynamics. The main achievement of our work is to develop a systematic
approach for constructing arbitrarily many correctors to the leading amplitude.
Based on a suitable duality formula, we exhibit necessary and sufficient
solvability conditions for the corrector equations that need to be solved
iteratively. Theverification of these solvability conditions is based on a
combination of mere algebra and arguments of combinatorial analysis. The
construction of arbitrarily many correctors enables us to produce infinitely
accurate approximate solutions to the free boundary problem. Eventually, we
show that the rectification phenomenon exhibited by Marcou in the context of
Rayleigh waves does not arise in the same way for the current vortex sheet
problem.
