Weakly nonlinear surface waves in magnetohydrodynamics. I

Abstract

This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the system of ideal incompressible magnetohydrodynamics. Current vortex sheets are piecewise smooth solutions that satisfy suitable jump conditions on the (free) discontinuity surface. In this article, we complete an earlier work by Alì and Hunter (Quart. Appl. Math. 61(3) (2003) 451–474) and construct approximate solutions at any arbitrarily large order of accuracy to the three-dimensional free boundary problem 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 which we consider here, their leading amplitude is governed by a nonlocal Hamilton–Jacobi type equation known as the ‘HIZ equation’ (standing for Hamilton–Il’insky–Zabolotskaya (J. Acoust. Soc. Amer. 97(2) (1995) 891–897)) 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. We exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on mere algebra and arguments of combinatorial analysis, namely a Leibniz type formula which we have not been able to find in the literature. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the current vortex sheet equations. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves (C. R. Math. Acad. Sci. Paris 349(23–24) (2011) 1239–1244) does not arise in the same way for the current vortex sheet problem.