The development of travelling waves in a simple isothermal chemical system - I. Quadratic autocatalysis with linear decay

The possibility of travelling reaction–diffusion waves developing 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 → C (rate k 2 b ) is examined. Two simple solutions are obtained first, namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the reactant B. Both of these indicate that the criterion for the existence of a travelling wave is that k 2 < k 1 a 0 , where a 0 is the initial concentration of reactant A. The equations governing the fully developed travelling waves are then discussed and it is shown that these possess a solution only if this criterion is satisfied, i. e. only if k = k 2 / k 1 a 0 < 1. Further properties of these waves are also established and, in particular, it is shown that the concentration of A increases monotonically from its fully reacted state at the rear of the wave to its unreacted state at the front, while the concentration of B has a single hump form. Numerical solutions of the full initial value problem are then obtained and these do confirm that travelling waves are possible only if k < 1 and suggest that, when this condition holds, these waves travel with the uniform speed v 0 = 2√ (1 – k ). This last result is established by a large time analysis of the full initial value problem that reveals that ahead of the reaction–diffusion front is a very weak diffusion-controlled region into which an exponentially small amount of B must diffuse before the reaction can be initiated. Finally, the behaviour of the travelling waves in the two asymptotic limits k → 0 and k → 1 are treated. In the first case the solution approaches that for the previously discussed k = 0 case on the length scale associated with the reaction–diffusion front, with the difference being seen on a much longer, O ( k –1 ), length scale. In the latter case we find that the concentration of A is 1 + O (1 – k ) and that of B is O ((1 – k ) 2 ), with the thickness of the reaction–diffusion front being of O ((1 – k ) ½ ).