Nonsharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion

Abstract

When degenerate nonlinear diffusion is introduced into the Fisher equation, giving u t = ( uu x ) x + u(1 − u), the travelling wave structure changes so that there is a sharp-front wave for one particular wave speed, with smooth-front waves for all faster speeds. The sharp-front solution has been studied by a number of previous authors; the present paper is concerned with the smooth-front waves. The authors use heuristic arguments to derive a relationship between initial data and the travelling wave speed to which this initial data evolves. The relationship compares very well with the results of numerical simulations. The authors go on to consider the form of smooth-front waves with speeds close to that of the sharp-front solution. Using singular perturbation theory, they derive an asymptotic approximation to the wave which gives valuable information about the structure of the smooth-front solutions.