Ahead of the Fisher-KPP front

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.