The development of travelling waves in a simple isothermal chemical system. IV. Quadratic autocatalysis with quadratic decay

Abstract

The initiation of travelling reaction-diffusion waves in the chemical system governed by the quadratic autocatalytic or branching reaction A + B → 2B (rate k 1 ab) coupled with the decay or termination step B + B → C (rate k 4 b 2 ) is discussed. The system is described by the non-dimensional parameter K - k 4 / k 1 and parameters representing the local initial input of B. It is shown that a travelling wave of permanent form will develop for all K (and no matter how small the initial input of B). Bounds on the solution of the initial-value problem are obtained as well as numerical integrations of the governing equations. The structure of the permanent form travelling waves that arise is discussed in some detail, as well as the asymptotic limits K → 0 and K → ∞. The behaviour of the solution for this problem is compared with solutions found previously for other related simple autocatalytic systems with autocatalyst decay.

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

“…We conclude the introduction with three remarks. The first remark is that the reaction terms f (u) in the above-mentioned models [22,3,21,19,26,27,29,4] are smooth and increasing in u, and f (0) = 0 and f (0) > 0. Hence f in these models satisfies the assumptions (A1)-(A3).…”

“…This modified version of the system (1.1) has been analyzed by Merkin and Needham [26,27,29], and their asymptotical analysis indicates the existence of the threshold concentration of the autocatalyst B for the generation of traveling waves. We remark that unlike the case with f (u) = k 1 u and K = 0, the wave profile for the reactant A is of front type and the one for the autocatalyst B is of pulse type due to the decay.…”

“…To overcome this controversy on the absence of the threshold concentration in the system (1.1) with f (u) = k 1 u and K = 0, Gray [11] has suggested that the autocatalyst cannot be indefinitely stable; instead, it should undergo subsequent change. Along this line, Merkin and Needham [26,27,29] have assumed that the autocatalyst B decays to the inert product C at a rate of order m,…”

Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models

“…We note that the reaction term f in the existing models [16,18,7,21,20,19] satisfies the assumptions (A1)-(A4). Hence our result establishes the existence and uniqueness of traveling wave solutions to the reaction-diffusion system associated with the reactions (1.2)-(1.3) (i.e., the system (1.1) with f (u) = k 1 u and K = k 2 ) for an isothermal autocatalytic chemical reaction of any order in which the autocatalyst is assumed to decay to the inert product at a rate of the same order.…”

“…But, the parameter m can be different from 1 in the context of higher order autocatalytic reactions. Specifically, we follow Merkin and Needham [21,20,19] to consider the general mth order isothermal autocatalytic chemical reaction where u and v are the concentration of the reactant A and the autocatalyst B, respectively, and the k i are the rate constants. Then by applying the law of mass action, the above reaction schemes lead to the system (1.1) with f (u) = k 1 u and K = k 2 .…”