Tracking Waves and Spiral Drift in Reaction−Diffusion Systems with Finite Bandwidth Dispersion Relations

Spiral defect drift in the wave fields of multiple excitation patterns

Spiral waves in excitable systems decay to drifting defects if forced by high-frequency wave trains. Using the Barkley model we analyze the drift velocity in planar wave trains as a function of wave frequency. Within two antiparallel, planar wave trains of equal frequency a defect is pushed into the collision region where it stops. Within two circular wave fields, however, it continues its drift in a direction perpendicular to the axis connecting the pacemakers. Depending on the forcing frequency and the initial position, this motion occurs either away from or toward the pacemaker axis. Three circular wave fields can be used to position the defect at a unique point close to the center of the pacemaker triangle. The results are also observed in experiments with the Belousov-Zhabotinsky reaction.

Wave-pinned filaments of scroll waves

Scroll waves are three-dimensional excitation patterns that rotate around one-dimensional space curves. Typically these filaments are closed loops or end at the system boundary. However, in excitable media with anomalous dispersion, filaments can be pinned to the wake of traveling wave pulses. This pinning is studied in experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction and a three-variable reaction-diffusion model. We show that wave-pinned filaments are related to the coexistence of rotating and translating wave defects in two dimensions. Filament pinning causes a continuous expansion of the total filament length. It can be ended by annihilating the pinning pulse in a frontal wave collision. Following such an annihilation, the filament connects itself to the system boundary. Its postannihilation shape that is initially the exposed rim of the scroll wave unwinds continuously over numerous rotation periods.

Scroll Wave Filaments Terminate in the Back of Traveling Fronts

Experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction demonstrate that three-dimensional scroll waves can rotate around filaments that end in the wake of a traveling excitation pulse. The vortex structures nucleate during the collision of three nonrotating excitation pulses. The nucleation process and the wave-termination of filaments are direct consequences of the system's anomalous dispersion relation. Vortex filaments are found to expand with about twice the speed of their anchoring wave fronts. Filament expansion is accompanied by the build-up of phase differences in spiral rotation creating strongly twisted wave structures. Experiments employ optical tomography for the reconstruction of the three-dimensional wave patterns.

Propagation failures, breathing pulses, and backfiring in an excitable reaction-diffusion system

We report results from experiments with a pseudo-one-dimensional Belousov-Zhabotinsky reaction that employs 1,4-cyclohexanedione as its organic substrate. This excitable system shows traveling oxidation pulses and pulse trains that can undergo complex sequences of propagation failures. Moreover, we present examples for (i) breathing pulses that undergo periodic changes in speed and size and (ii) backfiring pulses that near their back repeatedly generate new pulses propagating in opposite direction.

Propagation failure dynamics of wave trains in excitable systems

We report experimental and numerical results on temporal patterns of propagation failures in reaction-diffusion systems. Experiments employ the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagation failures occur in the frontier region of the wave train and can profoundly affect its expansion speed. The specific rhythms observed vary from simple periodic to highly complex and possibly chaotic sequences. All but the period-1 sequences are found in the transition region between "merging" and "tracking" dynamics, which correspond to wave behavior caused by two qualitatively different types of anomalous dispersion relations.

Dynamics of excitation pulses with attractive interaction: kinematic analysis and chemical wave experiments

We present a theoretical analysis of stacking and destacking wave trains in excitable reaction-diffusion systems with anomalous velocity-wavelength dependence. For linearized dispersion relations, kinematic analysis yields an analytical function that rigorously describes front trajectories. The corresponding accelerations have exactly one extremum that slowly decays with increasing pulse number. For subsequent pulses these maxima occur with a lag time equal to the inverse slope of the linearized dispersion curve. These findings are reproduced in experiments with chemical waves in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction but should be also applicable to step bunching on crystal surfaces and certain traffic phenomena.