Fronts and patterns in reaction-diffusion equations.

Abstract

This is a study of fronts and patterns formed in reaction-diffusion systems. A doubly-diffusive version of the two component FitzHugh-Nagumo equations with bistable reaction dynamics is investigated as an abstract model for the study of pattern phenomenologies found in many different physical systems. Front solutions connecting the two stable uniform states are found to be key building blocks for understanding extended patterns such as stationary domains and traveling pulses in one dimension, and labyrinthine structures, splitting spots, and spiral wave turbulence in two dimensions. The number and type of front solutions is controlled by a bifurcation that we derive both analytically and numerically. At this bifurcation, called the nonequilibrium Ising-Bloch (NIB) bifurcation, a single stationary Ising front loses stability to a pair of counterpropagating Bloch fronts. In two dimensions, we derive a boundary where extended fronts become unstable to transverse perturbations. In addition, near the NIB bifurcation, we discover a multivalued relation between the front speed and general perturbations such as curvature or an external convective field. This multivalued form allows perturbations to induce transitions that reverse the direction of front propagation. When occurring locally along an extended front, these transitions nucleate spiral-vortex pairs. The NIB bifurcation and transverse instability boundaries divide parameter space into regions of different pattern behaviors. Before the bifurcation, the system may form transient patterns or stationary domains consisting of pairs of Ising fronts. Above the transverse instability boundary, two-dimensional planar fronts destabilize, grow, and finger to form a space-filling labyrinthine, or lamellar, pattern. Beyond the bifurcation the multiplicity of Bloch front solutions allows for the formation of persistent traveling pulses and spiral waves. Near the NIB bifurcation there is an intermediate region where new unexpected patterns are found. One-dimensional stationary domains become unstable to oscillating or breathing domains. In two dimensions, the transverse instability and local front transitions are the mechanisms behind spot splitting and the development of spiral wave turbulence. Similar patterns have been observed recently in the ferrocyanide-iodate-sulfite reaction.