  The paper is concerned with the free boundary problem for 2D current-vortex
sheets in ideal incompressible magneto-hydrodynamics near the transition point
between the linearized stability and instability. In order to study the
dynamics of the discontinuity near the onset of the instability, Hunter and
Thoo have introduced an asymptotic quadratically nonlinear integro-differential
equation for the amplitude of small perturbations of the planar discontinuity.
We study such amplitude equation and prove its nonlinear well-posedness under a
stability condition given in terms of a longitudinal strain of the fluid along
the discontinuity.
