THE ECKHAUS INSTABILITY IN HEXAGONAL PATTERNS

The Eckhaus instability of hexagonal patterns is studied within the mo del of three coupled envelope equations for the underlying roll system s. The regions of instability in the parameter space are found analyti cally from both the phase approximation and a full system of amplitude equations. Beyond the stability limits of hexagons two different mode s go unstable. Both provide symmetry breaking of an initially regular pattern via splitting of a triplet of rolls into two triplets of growi ng disturbances. The parameters of fastest growing disturbances (wavel ength, orientation, growth rate) are determined from the full set of l inearized amplitude equations. The nonlinear stage of the Eckhaus inst ability is investigated numerically. Symmetry breaking due to the Eckh aus instability indeed occurs within a certain range of parameters, wh ich for small supercriticality parameter mu leads to a metastable diso rdered hexagonal state with numerous line and point defects. For large r mu the Eckhaus instability triggers the transition of regular hexago nal pattern to disordered roll state. The roll phase originates in the cores of defects and then spreads all over the pattern.