Eckhaus Instability in 2 Dimensions
One of the generic instabilities steady patterns exhibit is the
Eckhaus instability. It is a long-wave instability with respect to compression and expansion of the
pattern. It usually eventually leads to the descruction of one pair of vortices or rolls of the pattern.
In two-dimensional patterns it is not to be expected that the roll pair disappears at all locations at the
same time. Much more likely is the formation of defects. This process is shown in the figure below which
presents snapshots of the zero contour lines of the real and imaginary parts of the complex order parameter
describing the pattern near onset. At the intersection of these sets of lines the amplitude of the
pattern is identical zero. These points represent defects (dislocations) in the pattern. They are
always generated and destroyed in pairs with opposite "charge".
