Local amplitude equation from non-local dynamics

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 strongly non-local due to the long range character of the magnetic interaction. We develop a systematic multiple scale perturbation expansion in which the non-locality is reduced to the occurrence of the Hilbert transform of the surface profile. The resulting third order amplitude equation describing the slow modulation of the basic pattern is shown to be purely local.