The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves

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Abstract
We study the isothermal autocatalytic system , A + n B → ( n + 1)B , where n = 1 or 2 for quadratic or cubic autocatalysis respectively. In addition, we allow the chemical species, A and B, to have unequal diffusion rates. The propagating reaction-diffusion waves that may develop from a local initial input of the autocatalyst, B, are considered in one spatial dimension. We find that travelling wave solutions exist for all propagation speeds v ≥ v * n ,where v * n is a function of the ratio of the diffusion rates of the species A and B and represents the minimum propagation speed. It is also shown that the concentration of the autocatalyst, B, decays exponentially ahead of the wavefront for quadratic autocatalysis. However, for cubic autocatalysis, although the concentration of the autocatalyst decays exponentially ahead of the wavefront for travelling waves which propagate at speed v = v * 2 , this rate of decay is only algebraic for faster travelling waves with v > v * 2 . This difference in decay rate has implications for the selection of the long time wave speed when such travelling waves are generated from an initial-value problem.