The development of travelling waves in a simple isothermal chemical system II. Cubic autocatalysis with quadratic and linear decay

The possibility of travelling reaction-diffusion waves developing in the isothermal chemical system governed by the cubic autocatalytic reaction A + 2B → 3B (rate k 3 ab 2 ) coupled with either the linear decay step B → C (rate k 2 b ) or the quadratic decay step B + B → C (rate k 4 b 2 ) is examined. Two simple solutions are obtained,namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the autocatalyst B. Both of these suggest that, for the quadratic decay case, a wave will develop only if the non-dimensional parameter k ═ k 4 / k 3 a 0 < 1 (where a 0 is the initial concentration of the reactant A), with there being no restriction on the initial input of the autocatalyst B. However, for the linear decay case the initiation of a travelling wave depends on the parameter v ═ k 2 / k 3 a 2 0 and that, in addition, there is an input threshold on B before the formation of a wave will occur. The equations governing the fully developed travelling waves are then considered and it is shown that for the quadratic decay case the situation is similar to previous work in quadratic autocatalysis with linear decay, with a necessary condition for the existence of a travelling-wave solution being that K < 1. However, the case of linear decay is quite different, with a necessary condition for the existence of a travelling wave solution now found to be v < 1/4 Numerical solutions of the equations governing this case reveal further that a solution exists only for v < v c , with v c ≈ 0.0465, and that there are two branches of solution for 0 < v < v c . The behaviour of these lower branch solutions as v → 0 is discussed. The initial-value problem is then considered. For the quadratic decay case it is shown that the uniform state a ═ a 0 , b ═ 0 is globally asymptotically stable (i. e. a → a 0 , b → 0 uniformly for large times) for all k > 1. For the linear decay case it is shown that the development of a travelling wave requires β 0 > v (where β 0 is a measure of the initial input of B) for v < v c . These theoretical results are then complemented by numerical solutions of the initial-value problem for both cases, which confirm the various predictions of the theory. The behaviour of the solution of the equations governing the travelling waves is then discussed in the limits K → 0, v → 0 and K → 1. In the first case the solution approaches the solution for K ═ 0 (or v =0) on the length scale of the reaction-diffusion front, with there being a long tail region of length scale O ( K -1 ) (or O ( v -1 )) in which the autocatalyst B decays to zero. In the latter case we find that the concentration of reactant A is 1 + O [(1 - k )] and autocatalyst B is O[(1 - k 2 ] with the thickness of the reaction-diffusion front becoming large, of thickness O [(1- k ) -3/2 ].