The dynamics of vortex filaments in the complex Ginzburg-Landau equation

An asymptotic theory is developed for scroll waves in the complex Ginzburg-Landau equation. The theory is valid in the limit of small vortex filament curvature, torsion and phase twist and in the absence of filament-filament interaction. Explicit expressions for the filament velocity and phase evolution are found. The theoretical results are verified numerically in the case of circular untwisted vortex rings and for straight and sinusoidal filaments with phase twist. Numerical evidence for the reconnection of vortex filaments in the complex Ginzburg-Landau equation is shown. An estimate is given for the maximum intervortex separation beyond which coplanar filaments of locally opposite charge will not reconnect. This is done by balancing the motion of the filaments toward each other that would result if they were straight (a two-dimensional effect) with the opposing motion due to the filament curvature. The estimated vortex separation is in good agreement with numerical experiment.