The 1 : 2 spatial resonance on a hexagonal lattice in two-layered Rayleigh–Bénard problems

This paper investigates pattern formation resulting from the interaction between steady modes with wavenumbers in the ratio 1 : 2. A two-layered Rayleigh–Benard problem with infinite extent is examined; two layers are separated by a non-deformable and thin dividing plate. This physical set-up is known to provide an exact 1 : 2 resonance between critical modes. Restricting the wavevectors of interacting modes on a hexagonal lattice, we derive 12-dimensional amplitude equations up to cubic order. Among steady solutions of the equations, the stable ones are hexagons and mixed hexagons. Travelling waves corresponding to relative equilibria are also found to be stable depending on parameter values.