Precise asymptotics for Fisher-KPP fronts

Abstract

We consider the one-dimensional Fisher-KPP equation with step-like initial data. Nolen, Roquejoffre, and Ryzhik showed that the solution $u$ converges at long time to a traveling wave $\varphi$ at a position $\tilde{\sigma }(t)=2t-(3/2)\log t+{\alpha }_0-3\sqrt{\pi }/\sqrt{t}$, with error $O(t^{\gamma -1})$ for any $\gamma >0$. With their methods, we find a refined shift $\sigma (t)=\tilde{\sigma }(t)+{\mu }_{\ast }(\log t)/t+{\alpha }_1/t$ such that in the frame moving with $\sigma$, the solution $u$ satisfies $u(t,x)=\varphi (x)+\psi (x)/t+O(t^{\gamma -3/2})$ for a certain profile $\psi$ independent of initial data. The coefficient ${\alpha }_1$ depends on initial data, but ${\mu }_{\ast }=9(5-6\log 2)/8$ is universal, and agrees with a finding of Berestycki, Brunet, and Derrida in a closely-related problem. Furthermore, we predict the asymptotic forms of $\sigma$ and $u$ to arbitrarily high order.

38 pages