Spherically Symmetric Waves of a Reaction-Diffusion Equation.

Reaction-Diffusion Equations have been used to model nerve impulse propagation and spatially inhomogeneous situations in chemically reacting systems and population genetics. The kind of solutions that are often of interest in these areas are wave-like and do not die out. If the underlying spatial domain for the equation is one-dimensional, the description of such wave behaviour amounts to finding a travelling wave. This is a solution whose evolution in time under the equation is given by translating along the axis. A nerve impulse is an example of a travelling wave. If a wave in higher dimensions is expected, for instance one that is a function only of radius in a spherical co-ordinate system, this mathematical approach is not available. Such spherical waves look like one dimensional wave a long way out. In this report we invert this idea and deduce spherical wave behaviour in a model example where there is a stable one dimensional travelling wave. This approach also determines the remaining work that is necessary for a complete picture of the spherical wave behaviour for this equation.