Motion of Scroll Wave Filaments in the Complex Ginzburg-Landau Equation

Rotating spiral waves are observed in a variety of physical, chemical, and biological settings including the Belousov-Zhabotinsky (BZ) reaction, thermal convection in a thin fluid layer, slime mold on a nutrient-supplied medium, and waves of electrical activity in heart tissue [1,2]. While much attention has been devoted to spiral waves in two dimensions, there has also been increasing interest in the study of spiral waves in three dimensions or “scroll waves” [2]. The simple point singularity or “defect” at the center of a two-dimensional (2D) spiral wave now becomes a line defect known as the scroll wave filament which can be straight, curved, closed to form a loop, knotted, or interlinked with other loops. The scroll wave can be given a “twist” by allowing for a relative phase difference of the spirals along the filament. Scroll waves have been observed experimentally in the BZ reaction [3], in slime mold [4], and they are also believed to occur in the heart [5]. For a large class of extended systems in the vicinity of a Hopf bifurcation, expansion of the relevant equations [6] leads to a universal equation called the complex Ginzburg-Landau equation (CGLE),