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Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension

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Abstract

The possibility of initiating reaction-diffusion waves in an autocatalytic system represented schematically byA→B, ratekab p (p >- 1, witha, b being the concentrations ofA andB respectively) is considered through the local input ofB, measured by the parameter β0, into an otherwise uniform expanse ofA. It is shown that for 1 <-p < 1 + (2/N) (whereN is the space dimension) waves develop no matter how small the value of β0, while forp > 1 + (2/N) there is some threshold value of β0 below which waves are not formed, with diffusion playing the dominant role throughout. A lower bound for this threshold value is found. The permanent-form travelling wave equations are then discussed and the behaviour of the solution asp → 1 is considered in detail. It is shown that a three-region structure develops with the asymptotic wave speedv being singular (of the formv ∼ 2−2.3381 (p- 1)2/3) asp → 1.

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Authors and Affiliations

  1. Dept of Applied Mathematics, University of Leeds, LS2 9JT, Leeds

    J. H. Merkin

  2. School of Mathematics, University of East Anglia, NR4 7TJ, Norwich, UK

    D. J. Needham

Authors
  1. J. H. Merkin
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  2. D. J. Needham
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Merkin, J.H., Needham, D.J. Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension. Z. angew. Math. Phys. 44, 707–721 (1993). https://doi.org/10.1007/BF00948484

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  • DOI: https://doi.org/10.1007/BF00948484

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