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 convec- tion [1]. 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