Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system

Abstract

The initiation and propagation of reaction-diffusion travelling waves in two regions coupled together by the linear diffusive interchange of the autocatalytic species is considered via an initial-value problem in which amounts of the autocatalyst are introduced locally into otherwise uniform concentrations of the other species. The reaction in one region is given by quadratic autocatalysis, while the reaction in the other is given by quadratic autocatalysis together with the linear decay of the autocatalyst. A priori bounds for the initial-value problem are obtained first. These, together with the solution valid for small inputs of the autocatalyst, enable conditions to be derived under which travelling waves can be initiated giving a wave for all γ with k <2, or if k <(2γ−1)/(γ−1), where k and γ are dimensionless groups corresponding to the rate of chemical decay of the autocatalyst and to the strength of coupling respectively. The global asymptotic stability of the unreacted state is then discussed. A solution valid for strong coupling between the two regions is then derived. The equations governing the permanent-form travelling waves are treated in some detail, general properties of their solution and a solution valid for weak coupling being derived. Finally, the large-time solution of the initial-value problem is considered. This shows that, when travelling waves are initiated, they travel with their minimum possible speed: υ0=2[2−k−2γ+(k2+4γ2)1/2]1/2