Minimal model of binary fluid convection

Steady and oscillatory convection in a binary fluid mixture heated from below is considered. Stress-free and fixed temperature and concentration boundary conditions are used at the top and bottom, with periodic boundary conditions in the horizontal. A minimal Galerkin truncation is constructed such that the local branching behavior near both the steady state and Hopf bifurcations is correctly determined. To do this, modes generated through fourth order in perturbation theory must be retained. The properties of the steady states and standing, traveling, and modulated traveling waves in the resulting system are systematically analyzed. Broad agreement with the results of an analysis of a codimension-two bifurcation with O(2) symmetry is found.