The effect of the order of the autocatalysis on the transverse stability of reaction fronts

A linear stability analysis of planar reaction fronts to transverse perturbations is considered for a system based on an autocatalytic reaction of general order p. Dispersion curves, plots of the growth rate sigma against a transverse wavenumber k, are obtained for a range of values of p and D, where D is the ratio of the diffusion coefficients of autocatalyst and substrate. A value D(0) of D, dependent on p, is found at which sigma(max), the maximum value of sigma in the unstable regime, is largest, with D(0) increasing as p is increased. An asymptotic analysis for small wavenumbers is derived, which enables the region in the (p, D) parameter space for instability to be determined. An analysis for D small is undertaken, which leads to upper bounds on the wavenumber for a possible instability.