New stability results for patterns in a model of long-wavelength convection

We consider the transition from a spatially uniform state to a steady, spatially periodic pattern in a partial differential equation describing long-wavelength convection [E. Knobloch, Pattern selection in long-wavelength convection, Physica D 41 (1990) 450-479]. This both extends existing work on the study of rolls, squares and hexagons and demonstrates how recent generic results for the stability of spatially periodic patterns may be applied in practice. We find that squares, even if stable to roll perturbations, are often unstable when a wider class of perturbations is considered. We also find scenarios where transitions from hexagons to rectangles can occur. In some cases we find that, near onset, more exotic spatially periodic planforms are preferred over the usual rolls, squares and hexagons.