Vanishing corrections for the position in a linear model of FKPP fronts

TL;DR

This paper analyzes the asymptotic behavior of solutions to the linearized FKPP equation with absorbing boundary, identifying precise correction terms that govern convergence to a limiting profile, extending Bramson's and Ebert-van Saarloos' results.

Contribution

It provides the first-order identification of the correction term r(t) for the linear FKPP equation, extending known results to broader initial conditions and connecting linear and nonlinear cases.

Findings

For initial decay faster than x^ν e^{-x} with ν < -3, r(t) = -3√π/t.

For initial decay faster than x^ν e^{-x} with -3 ≤ ν < -2, correction modifies Bramson's log correction.

The correction term r(t) ensures the fastest convergence to the limiting profile ω(x).

Abstract

Take the linearised FKPP equation \[\partial_t h =\partial^2_x h +h\] with boundary condition $h(m(t),t)=0$. Depending on the behaviour of the initial condition $h_0(x)=h(x,0)$ we obtain the asymptotics - up to a $o(1)$ term $r(t)$ - of the absorbing boundary $m(t)$ such that $\omega(x):=\lim_t h(x+m(t) ,t)$ exists and is non-trivial. In particular, as in Bramson's results for the non-linear FKPP equation, we recover the celebrated $-(3/2)\log t$ correction for initial conditions decaying faster than $x^\nu e^{-x}$ for some $\nu<-2$. Furthermore, when we are in this regime, the main result of the present work is the identification (to first order) of the $r(t)$ term which ensures the fastest convergence to $\omega(x)$. When $h_0(x)$ decays faster than $x^\nu e^{-x}$ for some $\nu<-3$, we show that $r(t)$ must be chosen to be $-3\sqrt{\pi/t}$ which is precisely the term predicted heuristically by Ebert-van Saarloos in the non-linear case. When the initial condition decays as $x^\nu e^{-x}$ for some $\nu \in [-3,-2)$, we show that even though we are still in the regime where Bramson's correction is $-(3/2)\log t$, the Ebert-van Saarloos correction has to be modified. Similar results were recently obtained by Henderson using an analytical approach and only for compactly supported initial conditions.