Non-linear analysis of the Rosensweig instability

We derive a closed equation for the shape of the free surface of a magnetic fluid subject to an external magnetic field. The equation is non-local due to the long-range character of the magnetic interaction. We develop a systematic multiple-scale perturbation expansion with a non-local linear operator involving the Hilbert transform of the surface profile. The resulting amplitude equation describing the slow modulation of the basic pattern is shown to remain local