Convergence to a single wave in the Fisher-KPP equation

We study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as $2t-(3\mathrm{/}2)\log t+x_{\mathrm{\infty }}$, the solution of the equation converges as $t\to +\mathrm{\infty }$ to a translate of the traveling wave corresponding to the minimal speed~$c_{\ast }=2$. The constant $x_{\mathrm{\infty }}$ depends on the initial condition $u(0,x)$. The proof is elaborate, and based on probabilistic arguments. The purpose of this paper is to provide a simple proof based on PDE arguments.