Bifurcation on the hexagonal lattice and the planar Bénard problem

Abstract

One of the simplifications, used by Sattinger (1978), in studying the planar Bénard problem is to assume that the solutions are doubly periodic with respect to the hexagonal lattice in the plane. Once one makes this assumption, the generic situation is that the kernel of the linearized Boussinesq equations (linearized about the pure conduction solution) is six-dimensional, the eigenfunctions being superpositions of plane waves along three directions at mutual angles of 120°. In this situation the Liapunov-Schmidt procedure leads to a reduced bifurcation problem of the form g ( x, λ ) = 0 where g: [R6 x R -» R6 is smooth. Here λ represents the Rayleigh number. Moreover, such a g must commute with the symmetry group of the hexagonal lattice. In the paper we study such covariant bifurcation problems from the point of view of singularity theory and group theory, thus refining the work of Sattinger (1978). In particular we are able to classify the simplest such bifurcation problems as well as all of their perturbations. We find that stable rolls and stable hexagons occur as possible solutions. In addition, we find a rich structure of non-stable equilibrium solutions including wavy rolls and false hexagons appearing in the unfoldings of even the simplest degenerate bifurcation problems.