
TL;DR
This paper refines the long-term asymptotic analysis of Fisher-KPP equation solutions, providing precise correction terms for the wave front position and solution profile, revealing universal constants and high-order asymptotics.
Contribution
It introduces a refined shift for the wave front and detailed asymptotic expansions, including universal constants, improving understanding of Fisher-KPP front dynamics.
Findings
Universal constant (5-6 log 2)/8 in wave speed correction
Precise asymptotic expansion of solution profile with order t^{-1}
Prediction of high-order asymptotic forms for wave position and solution
Abstract
We consider the one-dimensional Fisher-KPP equation with step-like initial data. Nolen, Roquejoffre, and Ryzhik showed that the solution converges at long time to a traveling wave at a position , with error for any . With their methods, we find a refined shift such that in the frame moving with , the solution satisfies for a certain profile independent of initial data. The coefficient depends on initial data, but is universal, and agrees with a finding of Berestycki, Brunet, and Derrida in a closely-related problem. Furthermore, we predict the asymptotic forms of and to arbitrarily high order.
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