The Evolution of Travelling Waves in Fractional Order Autocatalysis with Decay. I. Permanent Form Travelling Waves

This paper considers a model, isothermal, autocatalytic reaction scheme with termination when the reaction orders m,n are fractional, i.e., 0 n and k∈(0,kc), where k c depends upon m and n. It is further established that these wavefronts are of excitable (rather than Fisher--Kolmogorov) type and have only semi-infinite support (that is, the reaction is totally dormant ahead of the wavefront). These features are in contrast to the case when m,n ≥ 1 as studied by Merkin, Needham, and Scott [Proc. Roy. Soc. London Ser. A, 424 (1989), pp. 187--209], Merkin and Needham [ J. Engrg. Math., 23 (1989), pp. 343--356], and Needham and Merkin [ Phil Trans. Roy. Soc. London Ser. A, 337 (1991), pp. 261--274]. It should be noted that the corresponding reaction--diffusion equations are singular in the sense that the reaction and decay terms are not Lipschitz continuous at low concentration and that this requires modifications to the standard travelling wave theory for regular reaction--diffusion equations (see, for example, Fife [ Mathematical Aspects of Reacting and Diffusing Systems Lecture Notes in Biomath. 28, Springer-Verlag, New York, 1979]).