The Eckhaus instability in hexagonal patterns

The Eckhaus instability of hexagonal patterns is studied within the model of three coupled envelope equations for the underlying roll systems. The regions of instability in the parameter space are found analytically from both the phase approximation and a full system of amplitude equations. Beyond the stability limits of hexagons two different modes go unstable. Both provide symmetry breaking of an initially regular pattern via splitting of a triplet of rolls into two triplets of growing disturbances. The parameters of fastest growing disturbances (wavelength, orientation, growth rate) are determined from the full set of linearized amplitude equations. The nonlinear stage of the Eckhaus instability is investigated numerically. Symmetry breaking due to the Eckhaus instability indeed occurs within a certain range of parameters, which for small supercriticality parameter p leads to a metastable disordered hexagonal state with numerous line and point defects. For larger/t the Eckhaus instability triggers the transition of regular hexagonal pattern to disordered roll state. The roll phase originates in the cores of defects and then spreads all over the pattern.