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A note of the convergence of the Fisher-KPP front centred around its $α$-level

We consider the solution $u(x,t)$ of the Fisher-KPP equation $\partial_t u=\partial_x^2u+u-u^2$ centred around its $α$-level $μ_t^{(α)}$ defined as $u(μ_t^{(α)},t)=α$. It is well known that for an initial datum that decreases fast enough, then $u(μ_t^{(α)}+x,t)$ converges as $t\to\infty$ to the critical travelling wave. We study in this paper the speed of this convergence and the asymptotic expansion of $μ_t^{(α)}$ for large~$t$. It is known from Bramson that for initial conditions that decay fast enough, one has $μ_t^{(α)}=2t-(3/2)\ln t+\text{Cste}+o(1)$. Work is under way \cite{nrr} to show that the $o(1)$ in the expansion is in fact a $k^{(α)}/\sqrt t+\mathcal O(t^{ε-1})$ for any $ε>0$ for some $k^{(α)}$, where it is not clear at this point whether $k^{(α)}$ depends or not on $α$. We show that, unless the time derivative of $μ_t^{(α)}$ has a very unexpected behaviour at infinity, the coefficient $k^{(α)}$ does not, in fact, depend on $α$. We also conjecture that, for an initial condition that decays fast enough, one has in fact $μ_t^{(α)}=2t-(3/2)\ln t+\text{Cste}-(3\sqrtπ)/\sqrt t+g (\ln t)/t +o (1/t)$ for some constant~$g$ which does not depend on $α$.