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+2B→3BA+2B→3B involving two chemical species AA and BB. Their diffusion coefficients, denoted by DADA and DBDB, 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, BB, are investigated in one spatial dimension. We show the existence of travelling wave solutions for all propagation speed v≥v2∗, with v2∗ a function of the ratio of the diffusion rates of the species AA and BB, 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≤v2v≤v2, where v2v2 depends on the ratio of the diffusion rates of the species AA and BB. 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+nB→(n+1)BA+nB→(n+1)B, of nn-th order with n>1n>1.