Instabilities in propagating reaction-diffusion fronts

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Abstract
Simple reaction‐diffusion fronts are examined in one and two dimensions. In one‐dimensional configurations, fronts arising from either quadratic or cubic autocatalysis typically choose the minimum allowable velocity from an infinite spectrum of possible wave speeds. These speeds depend on both the diffusion coefficient of the autocatalytic species and the pseudo‐first‐order rate constant for the autocatalytic reaction. In the mixed‐order case, where both quadratic and cubic channels contribute, the wave speed depends on the rate constants for both channels, provided the cubic channel dominates. Wave propagation is completely determined by the quadratic contribution when it is more heavily weighted. In two‐dimensional configurations, with unequal diffusion coefficients, the corresponding two‐variable planar fronts may become unstable to perturbations. The instability occurs when the ratio of the diffusion coefficient for the reactant to that for the autocatalyst exceeds some critical value. This critical value, in turn, depends on the relative weights of the quadratic and cubic contributions to the overall kinetics. The spatiotemporal form of the nonplanar wave in such systems depends on the width of the reaction zone, and a sequence showing Hopf, symmetry‐breaking, and period‐doubling bifurcations leading to chaotic behavior is observed as the width is increased.