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

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.

30 pages

References in corpus (3)

• Phenomenological picture of fluctuations in branching random walks
• The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach
• An exactly solvable travelling wave equation in the Fisher-KPP class

Cited by in corpus (6)

• Locality, Quantum Fluctuations, and Scrambling
• Exact solution and precise asymptotics of a Fisher-KPP type front
• A new approach to computing the asymptotics of the position of Fisher-KPP fronts
• Kinetics of information scrambling in correlated electrons: disorder-driven transition from shock-wave to FKPP dynamics
• Cross-overs of Bramson's shift at the transition between pulled and pushed fronts
• Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree