Instabilities of Hexagonal Patterns with Broken Chiral Symmetry

Three coupled Ginzburg-Landau equations for hexagonal patterns with broken chiral symmetry are investigated. They are relevant for the dynamics close to onset of rotating non-Boussinesq or surface-tension-driven convection. Steady and oscillatory, long- and short-wave instabilities of the hexagons are found. For the long-wave behavior coupled phase equations are derived. Numerical simulations of the Ginzburg-Landau equations indicate bistability between spatio-temporally chaotic patterns and stable steady hexagons. The chaotic state can, however, not be described properly with the Ginzburg-Landau equations.