Algebraic study of drifting spiral waves

This two-dimensional study is motivated by cardiac electrophysiology, and focuses on rotating spiral waves in reaction-diffusion (RD) models. Here we deal with a spiral's translational drift under a constant externally imposed gradient $\mathbf{G}.$ A long-standing problem may be stated as follows: Given the dimensionless drift velocity $\mathbf{V}/G,$ find its nontrivial direction angle $\mathrm{\Gamma }$ relative to $\mathbf{G}$. A deductive algebraic treatment yields a solution, $cos\mathrm{\Gamma }=-V/G.$ Three features are worth noting: the combination of algebraic and RD contexts; a somewhat extensive derivation contrasting with a compact result; and the generality due to the absence of reaction details in the formula. Agreement with a computational database is good to fair, if spirals of very low density are excluded.

Figure 1

Theory vs simulation data: the dimensionless drift speed $V/G$ is plotted against the drift angle $\mathrm{\Gamma }.$ The curve follows Eq. (30); the points are taken from the database of Ref. [4]. Comments are to be found in the text.