O(2)-symmetric bifurcation theory for convection rolls

The interaction of two steady state modes for the two-dimensional Benard problem with lateral periodic boundary conditions (p.b.c.) and free horizontal boundary conditions is considered. The p.b.c. generate an action of O(2) on the Navier-Stokes equations. The Boussinesq approximation induces an additional Z(2)-symmetry. The O(2) × Z(2) equivariant bifurcation equations are determined and two fifth order terms are identified. Two types of mixed mode solutions and a travelling wave solution are found. Using MACSYMA, the fifth order terms are calculated and the stability of the solutions to phase perturbations depending on the Prandtl number is determined.