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

Abstract

Explicit asymptotic analytical results are derived for the motion of scroll wave filaments in the complex Ginzburg-Landau equation. Good agreement with numerical tests is obtained. The analysis highlights the necessity of allowing for previously ignored small wave-number shifts in the propagation of the waves away from the filament. [S0031-9007(97) PACS numbers: 82.40.Ck, 47.32.Cc 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),where A is a complex scalar field, and a and b are real parameters. (For example, in a system of diffusing chemically reacting constituents, a nonzero b arises due to unequal diffusion coefficients of the chemicals.) Equation (1) exhibits spiral waves in two dimensions and scroll waves in three dimensions. In this paper, we shall study the fundamental problem of obtaining the dynamical behavior of a three-dimensional (3D) CGLE scroll wave filament [7].In a pioneering work by Keener [8], analysis techniques and results for the evolution of scroll wave filaments for general systems of reaction-diffusion equations were formulated. The results are based on certain hypotheses concerning the asymptotic form of the solution and the number of solutions of an adjoint equation that results from the analysis. Quantitative tests of the theory against numerical or laboratory experiments have never been made in the generic case of unequal diffusion coefficients [analogous to b fi 0 in (1)]. This is because coefficients in Keener's general theory are expressed in terms of inner products with solutions of the adjoint equation, which has never been solved. In our paper, by focusing on the simpler CGLE, we are able to derive explicit results and to quantitatively test them numerically. Our solution, while adopting some of Keener's techniques, also makes evident a defect in the hypotheses utilized in the previous theory. In particular, we find that it is necessary to include the possibility of w...

Dynamics of vortex lines in the three-dimensional complex Ginzburg-Landau equation: Instability, stretching, entanglement, and helices

“…In addition, there is no (at least, in the first order in 1/R) overall drift of the vortex ring in the direction perpendicular to the collapse motion. The collapse rate (called often "line tension") ν = 1 + b 2 appears to be in a reasonable agreement with the simulations [22]. This result generalises Keener's ansatz by including the curvatureinduced shift of the filament's wavenumber.…”

“…Recently, the dynamics of three-dimensional (3D) vortex lines in the CGLE has attracted substantial attention [21][22][23]. As a definition of a vortex line we accept a line singularity of the phase of a complex function A. Gabbay Ott and Guzdar [22] applied a generalisation of Keener's method for a scroll vortex in reaction-diffusion systems [20]. They derived that the ring of a radius R collapses in finite time according to the following evolution law…”

Phase chaos in the anisotropic complex Ginzburg-Landau equation

“…For a general review see e.g [6].The one-dimensional (1D) and the 2D isotropic cases have been investigated rather well [7]- [16]. A number of results have also been obtained in 3D [17,18]. Taking up some earlier work [19] we recently reported about spirals and ordered defect chains in the anisotropic complex Ginzburg-Landau equation(ACGLE) [20]Here A is the complex amplitude modulating the critical mode in space and time.…”

Twist localizes three-dimensional patterns

“…If there is a linear selection mechanism for large obliqueness angles at threshold, at least one of the tree conditions (3,8,9) must be invalid. The reason may be found in the truncation errors of the asymptotic expansion which leads to Eqs.…”

“…The reason may be found in the truncation errors of the asymptotic expansion which leads to Eqs. (3,8,9), the excluded effect of noise, or an insufficient description on the hydrodynamic level. All possible truncation errors become, by one way or another, large only for z 0 = O(d).…”