Ahead of the Fisher-KPP front

TL;DR

This paper analyzes the asymptotic behavior of solutions to the Fisher-KPP equation ahead of the propagating front, especially for velocities near the critical speed, revealing explicit and irregular prefactors.

Contribution

It provides an explicit asymptotic expansion of the solution ahead of the front for velocities close to 2, using a novel 'magical expression' linking key quantities.

Findings

Prefactor depends non-trivially on initial conditions and velocity

Asymptotic expansion is surprisingly explicit and irregular

Results apply to velocities c > 2 close to the critical speed

Abstract

The solution h to the Fisher-KPP equation with a steep enough initial condition develops into a front moving at velocity 2, with logarithmic corrections to its position. In this paper we investigate the value h(c t, t) of the solution ahead of the front, at time t and position c t, with c > 2. That value goes to zero exponentially fast with time, with a well-known rate, but the prefactor depends in a non-trivial way of c, the initial condition and the non-linearity in the equation. We compute an asymptotic expansion of that prefactor for velocities c close to 2. The expansion is surprisingly explicit and irregular. The main tool of this paper is the so-called "magical expression" which relates the position of the front, the initial condition, and the quantity we investigate.