Envelope Equations Near the Onset of a Hexagonal Pattern

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Abstract
We present the amplitude equation including slow spatial modulations for the hexagonal patterns observed near the onset of the Bénard instability in the presence of non-Boussinesq effects. In contrast to the onset of convection in Boussinesq approximation for rigid-rigid boundary conditions, we do not find a generalized thermodynamic potential. The same conclusion is found to hold for surface tension driven Marangoni convection and for temporally modulated convection. We also point out the applicability of our approach to other systems such as the Rosenzweig instability in ferrofluids as well as to the baroclinic instability and to the buckling of plates and shells, for which an envelope equation, which is second order in time, results.