On the transition from initial data to travelling waves in the Fisher-KPP equation

The Fisher-KPP equation u_t = u_xx + u(1 - u) has a travelling wave solution for all speeds = 2. Initial data that decrease monotonically from 1 to 0 on - 8 < x < 8, with u(x, 0) = O_s(e^-?x) as x ? 8, are known to evolve to a travelling wave, whose speed depends on ?. Here, it is shown that the relationship between wave speed and ? can be recovered by linearizing the Fisher-KPP equation about u = 0 and explicitly solving the linear equation. Moreover, the calculation predicts that in the case ? > 1, the solution for u_x/u itself evolves to a transition wave, moving ahead of the (minimum speed) u wave at the greater speed of 2?. Behind this transition, u_x/u = -x/(2t), while ahead of it, u_x/u = -?. The paper goes on to discuss the potential applications of the method to systems of coupled reaction-diffusion equations.