The development of travelling waves in cubic auto-catalysis with different rates of diffusion

In this paper we study the isothermal auto-catalytic chemical reaction, A + 2 B → 3 B involving two chemical species A and B . Their diffusion coefficients, denoted by D A and D B , respectively, are unequal, which happens when the two chemical species have different molecular weights. The propagating reaction–diffusion waves that may develop from a local initial input of the auto-catalyst, B , are investigated in one spatial dimension. We show the existence of travelling wave solutions for all propagation speed v ≥ v 2 ∗ , with v 2 ∗ a function of the ratio of the diffusion rates of the species A and B , representing the estimated minimum propagation speed. Our result improves significantly on the results of early works. In addition, we show the non-existence of travelling wave solutions when v ≤ v 2 , where v 2 depends on the ratio of the diffusion rates of the species A and B . We believe that our non-existence result is the first of the kind for equations of the isothermal auto-catalytic chemical reaction type. We also demonstrate similar results on the general isothermal auto-catalytic chemical reaction, A + n B → ( n + 1 ) B , of n -th order with n > 1 .