The adjoint problem in the presence of a deformed surface: the example of the Rosensweig instability on magnetic fluids

The Rosensweig instability is the phenomenon that above a certain threshold of a vertical magnetic field peaks appear on the free surface of a horizontal layer of magnetic fluid. In contrast to almost all classical hydrodynamical systems, the nonlinearities of the Rosensweig instability are entirely triggered by the properties of a deformed and a priori unknown surface. The resulting problems in defining an adjoint operator for such nonlinearities are illustrated. The implications concerning amplitude equations for pattern forming systems with a deformed surface are discussed.Comment: 11 pages, 1 figur

Fronts between hexagons and squares in a generalized Swift-Hohemberg equation

Pinning effects in domain walls separating different orientations in patterns in nonequilibrium systems are studied. Usually, theoretical studies consider perfect structures, but in experiments, point defects, grain boundaries, etc., always appear. The aim of this paper is to perform an analysis of the stability of fronts between hexagons and squares in a generalized Swift-Hohenberg model equation. We focus the analysis on pinned fronts between domains with different symmetries by using amplitude equations and by considering the small-scale structure in the pattern. The conditions for pinning effects and stable fronts are determined. This study is completed with direct simulations of the generalized Swift-Hohenberg equation. The results agree qualitatively with recent observations in convection and in ferrofluid instabilities

Fronts between hexagons and squares in a generalized Swift-Hohemberg equation

Pinning effects in domain walls separating different orientations in patterns in nonequilibrium systems are studied. Usually, theoretical studies consider perfect structures, but in experiments, point defects, grain boundaries, etc., always appear. The aim of this paper is to perform an analysis of the stability of fronts between hexagons and squares in a generalized Swift-Hohenberg model equation. We focus the analysis on pinned fronts between domains with different symmetries by using amplitude equations and by considering the small-scale structure in the pattern. The conditions for pinning effects and stable fronts are determined. This study is completed with direct simulations of the generalized Swift-Hohenberg equation. The results agree qualitatively with recent observations in convection and in ferrofluid instabilities