Refined long-time asymptotics for Fisher–KPP fronts

Abstract

We study the one-dimensional Fisher–KPP equation, with an initial condition [Formula: see text] that coincides with the step function except on a compact set. A well-known result of Bramson in [Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978) 531–581; Convergence of Solutions of the Kolmogorov Equation to Travelling Waves (American Mathematical Society, Providence, RI, 1983)] states that, as [Formula: see text], the solution converges to a traveling wave located at the position [Formula: see text], with the shift [Formula: see text] that depends on [Formula: see text]. Ebert and Van Saarloos have formally derived in [Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Phys. D 146 (2000) 1–99; Front propagation into unstable states, Phys. Rep. 386 (2003) 29–222] a correction to the Bramson shift, arguing that [Formula: see text]. Here, we prove that this result does hold, with an error term of the size [Formula: see text], for any [Formula: see text]. The interesting aspect of this asymptotics is that the coefficient in front of the [Formula: see text]-term does not depend on [Formula: see text].