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The Rosensweig instability has a special character among the frequentlydiscussed instabilities. One distinct property is the necessary presence of adeformable surface, and another very important fact is, that the driving forceacts purely via the surface and shows no bulk effect. These properties make itrather difficult to give a correct weakly nonlinear analysis. In this paper wegive a detailed derivation of the appropriate amplitude equation based on thehydrodynamic equations emphasizing the conceptually new procedures necessary todeal with the distinct properties mentioned above. First the deformable surfacerequires a fully dynamic treatment of the instability and the observedstationary case can be interpreted as the limiting case of a frozen-incharacteristic mode. Second, the fact that the driving force is manifest in theboundary conditions, only, requires a considerable change in the formalism ofweakly nonlinear bifurcation theory. To obtain the amplitude equations acombination of solubility conditions and (normal stress) boundary conditionshas to be invoked in all orders of the expansions.

The Amplitude Equation for the Rosensweig Instability in Magnetic Fluids and Gels