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Precise asymptotics for Fisher-KPP fronts
Analysis of PDEs · Submitted 7 Dec 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim With their methods, we find a refined shift $\sigma(t) = \tilde \sigma(t) + \mu_* (\log t)/t + \alpha_1/t$ such that in the frame moving with $\sigma$, the solution $u$ satisfies $u(t,x) = \phi (x) + \psi(x)/t + O(t^{\gamma-3/2})$ for a certain profile $\psi$ independent of initial data.
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The abstract, sentence by sentence
We consider the one-dimensional Fisher-KPP equation with step-like initial data. Nolen, Roquejoffre, and Ryzhik showed that the solution $u$ converges at long time to a traveling wave $\phi$ at a position $\tilde \sigma(t) = 2t - (3/2)\log t + \alpha_0- 3\sqrt{\pi}/\sqrt{t}$, with error $O(t^{\gamma-1})$ for any $\gamma>0$. With their methods, we find a refined shift $\sigma(t) = \tilde \sigma(t) + \mu_* (\log t)/t + \alpha_1/t$ such that in the frame moving with $\sigma$, the solution $u$ satisfies $u(t,x) = \phi (x) + \psi(x)/t + O(t^{\gamma-3/2})$ for a certain profile $\psi$ independent of initial data. The coefficient $\alpha_1$ depends on initial data, but $\mu_* = 9(5-6\log 2)/8$ is universal, and agrees with a finding of Berestycki, Brunet, and Derrida in a closely-related problem. Furthermore, we predict the asymptotic forms of $\sigma$ and $u$ to arbitrarily high order.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the one-dimensional Fisher-KPP equation with step-like initial data. | background 0.92 | 0.38 | 0.02 | asserted 0.80 |
| s1 | Nolen, Roquejoffre, and Ryzhik showed that the solution $u$ converges at long time to a traveling wave $\phi$ at a position $\tilde \sigma(t) = 2t - (3/2)\log t + \alpha_0- 3\sqrt{\pi}/\sqrt{t}$, with error $O(t^{\gamma-1})$ for any $\gamma>0$. | background 1.00 | 0.03 | 0.07 | asserted 1.00 |
| s2 | With their methods, we find a refined shift $\sigma(t) = \tilde \sigma(t) + \mu_* (\log t)/t + \alpha_1/t$ such that in the frame moving with $\sigma$, the solution $u$ satisfies $u(t,x) = \phi (x) + \psi(x)/t + O(t^{\gamma-3/2})$ for a certain profile $\psi$ independent of initial data. | result 0.82 | 0.95 | 0.70 | asserted 1.00 |
| s3 | The coefficient $\alpha_1$ depends on initial data, but $\mu_* = 9(5-6\log 2)/8$ is universal, and agrees with a finding of Berestycki, Brunet, and Derrida in a closely-related problem. | result 0.97 | 0.75 | 0.23 | asserted 1.00 |
| s4 | Furthermore, we predict the asymptotic forms of $\sigma$ and $u$ to arbitrarily high order. | result 0.60 | 0.84 | 0.16 | asserted 0.59 |
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Original abstract
We consider the one-dimensional Fisher-KPP equation with step-like initial data. Nolen, Roquejoffre, and Ryzhik showed that the solution $u$ converges at long time to a traveling wave $\phi$ at a position $\tilde \sigma(t) = 2t - (3/2)\log t + \alpha_0- 3\sqrt{\pi}/\sqrt{t}$, with error $O(t^{\gamma-1})$ for any $\gamma>0$. With their methods, we find a refined shift $\sigma(t) = \tilde \sigma(t) + \mu_* (\log t)/t + \alpha_1/t$ such that in the frame moving with $\sigma$, the solution $u$ satisfies $u(t,x) = \phi (x) + \psi(x)/t + O(t^{\gamma-3/2})$ for a certain profile $\psi$ independent of initial data. The coefficient $\alpha_1$ depends on initial data, but $\mu_* = 9(5-6\log 2)/8$ is universal, and agrees with a finding of Berestycki, Brunet, and Derrida in a closely-related problem. Furthermore, we predict the asymptotic forms of $\sigma$ and $u$ to arbitrarily high order.
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References
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3 arXiv papers cite this in 3 sentences
Papers citing it
Cites it as background 3 papers
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The determination of the position $μ_t$ of the Fisher-KPP equation has attracted an uninterrupted attention [ref] since the equation was introduced in 1937 [ref] .
A new approach to computing the asymptotics of the position of Fisher-KPP fronts · Introduction · 1.00
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More precise results about the fine asymptotics of the front location were obtained in [ref] .
Branching Brownian motion with decay of mass and the non-local Fisher-KPP equation · The non-local Fisher-KPP equation · 0.80
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We note that a rich literature exists on $1/√(t)$ corrections for solutions to the F-KPP equation and related equations, see e.g. [ref] and the references therein.
1-stable fluctuations in branching Brownian motion at critical temperature I: the derivative martingale · Comments and heuristics · 0.98
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