Reaction-diffusion and phase waves occurring in a class of scalar reaction-diffusion equations

In a recent paper Sherratt and Marchant (1946 IMA J. Appl. Math. 56 289-302) have made numerical studies of initial value problems for the cubic Fisher equation. It was observed that for initial data with sufficiently slow decay rate at large x (dimensionless distance), the long-time evolution involved the development of an accelerating and stretching wavefront, rather than the more usual permanent form, constant velocity reaction-diffusion wavefront. They also presented a heuristic argument to obtained an expression for the propagation speed of this wavefront. In this paper we address the same initial value problem for the generalized Fisher equation of order m greater than or equal to 1. Using matched asymptotic expansions we are able to obtain the complete structure of the solution to the initial value problem for large time. This confirms the computations of Sherratt and Marchant for m > I,and gives complete information about the structure and propagation speed of the evolving, accelerating, phase wave. In addition, we also find that phase waves can propagate in the case m = I, which is contrary to the suggestions of Sherratt and Marchant. For m = 1, the asymptotics are exponential in I (dimensionless time), whilst for m > 1 the asymptotics are algebraic in I.