On the transition from initial data to travelling waves in the fisher-kpp equation

Abstract

The Fisher-KPP equation has a travelling wave solution for all speeds . Initial data that decrease monotonically from 1 to 0 on , with as , 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 and explicitly solving the linear equation. Moreover, the calculation predicts that in the case , the solution for itself evolves to a transition wave, moving ahead of the (minimum speed) u wave at the greater speed of . Behind this transition , while ahead of it . The paper goes on to discuss the potential applications of the method to systems of coupled reaction-diffusion equations