Precise asymptotics for Fisher-KPP fronts

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 $\tilde \sigma(t) = 2t - (3/2)\log t + \alpha_0- 3\sqrt{\pi}/\sqrt{t}$,witherror$O(t^{\gamma-1})$ for any $\gamma >0$. With their methods, we find a refined shift $\sigma(t) = \tilde \sigma(t) + \mu_* (\log t)/t + \alpha_1/t$suchthatintheframemovingwith$\sigma$,thesolution$u$satisfies$u(t,x) = \phi (x) + \psi(x)/t + O(t^{\gamma-3/2})$foracertainprofile$\psi$ independent of initial data. The coefficient ${\alpha }_1$ depends on initial data, but $\mu_* = 9(5-6\log 2)/8$ 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.