The structure of spiral-domain patterns and shocks in the 2D complex Ginzburg-Landau equation

Cellular patterns appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field. In the case of the complex Ginzburg-Landau equation, disordered cells of effectively frozen spirals appear, separated by thin walls (shocks), on a scale much larger than the basic wavelength of the spirals, We show that these structures can be understood in very simple terms. In particular, we show that the walls are, to a good approximation, segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex, The fact that the phase is only defined up to an integer multiple of 2 pi introduces a quantization condition on the sizes of the smallest spiral domains. The transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations. The structure depends on the angle the wall makes with the local phase contours, and the behavior can be either monotonic or oscillatory, depending on the parameters.