Precise asymptotics for Fisher–KPP fronts

Abstract

We consider the one-dimensional Fisher–KPP equation with step-like initial data. Nolen et al showed that the solution u converges at long time to a travelling wave at a position , with error for any . With their methods, we find a refined shift such that in a frame moving with , the solution u satisfies for a certain profile independent of the initial data. The coefficient depends on the initial data, but is universal, and agrees with a finding of Berestycki et al. Furthermore, we predict the asymptotic forms of and u to arbitrarily high order.