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# The Development of Travelling Waves in a Simple Isothermal Chemical System III. Cubic and Mixed Autocatalysis

P. Gray, J. H. Merkin, D. J. Needham and S. K. Scott
Proceedings: Mathematical and Physical Sciences
Vol. 430, No. 1880 (Sep. 8, 1990), pp. 509-524
Stable URL: http://www.jstor.org/stable/79953
Page Count: 16
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## Abstract

Autocatalytic chemical reactions can support isothermal travelling waves of constant speed and form. This paper extends previous studies to cubic autocatalysis and to mixed systems where quadratic and cubic autocatalyses occur concurrently. \aligned \text{A}+\text{B}\rightarrow 2\text{B},\quad & \text{rate}=k_{\text{q}}\,ab,\quad \quad \,(1)\\ \text{A}+2\text{B}\rightarrow 3\text{B},\quad & \text{rate}=k_{\text{c}}\,ab^{2}.\quad \quad (2) \endaligned For pure cubic autocatalysis the wave has, at large times, a constant asymptotic speed v0 (where v0 = $\frac{1}{\sqrt{2}}$ in the appropriate dimensionless units). This result is confirmed by numerical investigation of the initial-value problem. Perturbations to this stable wave-speed decay at long times as t-3/2e-1/8t. The mixed system is governed by a non-dimensional parameter μ = kq/kca0 which measures the relative rates of transformation by quadratic and cubic modes. In the mixed case (μ ≠ 0) the reaction-diffusion wave has a form appropriate to a purely cubic autocatalysis so long as μ lies between 1/2 and 0. When μ exceeds 1/2, the reaction wave loses its symmetrical form, and all its properties steadily approach those of quadratic autocatalysis. The value μ =1/2 is the value at which rates of conversion by the two paths are equal.

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