Modulated two-dimensional patterns in reaction-diffusion systems

New modulation equations for hexagonal patterns in reaction-diffusion systems are derived for parameter regimes corresponding to the onset of patterns. These systems include additional nonlinearities which are not present in Rayleigh-Benard convection or Swift-Hohenberg type models. The dynamics of hexagonal and roll patterns are studied using a combination of analytical and computational approaches which exploit the hexagonal structure of the modulation equations. The investigation demonstrates instabilities and new phenomena not found in other systems, and is applied to patterns of flame fronts in a certain model of burner stabilized flames.