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. Introduction. – The emergence of spatio-temporal order in distributed systems can often be theoretically analyzed in terms of amplitude equations [1, 2]. Generically these amplitude equations are non-linear partial differential equations for the slow time and space variations of an envelope function of unstable modes. As such they reflect the local character of the underlying dynamics. In several interesting cases, however, the dynamics of the system is substantially influenced by non-local interactions. These may arise, e.g., from a mean flow in convection problems [3, 4, 5, 6], from electric and magnetic fields in solids [7, 8] or electrochemical systems [9], from long range elastic interactions [10], or from global couplings in systems with chemical reactions [11, 12, 13]. Since the type and stability of the emerging patterns is usually modified if non-local interactions are present the corresponding amplitude equations are expected to exhibit some degree of non-locality as well. From a phenomenological point of view one may therefore be tempted to simply add to the standard form of an amplitude equation non-local terms complying with the relevant symmetries of the system [14,15]. A systematic derivation of the amplitude equation from the basic non-local dynamics has been accomplished in a few cases only. [7, 11, 6, 10, 13]. In the present letter we investigate the formation of static patterns on the free surface of a magnetic fluid subject to an external magnetic field. In this process the non-local character of the magnetic interaction is of vital importance. We establish a closed, strongly non-local equation for the free surface of the magnetic fluid and systematically derive an amplitude equation for the surface deflection in the vicinity of the critical field strength. Although the