{
  "id": "1712.02472",
  "version": "v1",
  "published": "2017-12-07T02:08:58.000Z",
  "updated": "2017-12-07T02:08:58.000Z",
  "title": "Precise asymptotics for Fisher-KPP fronts",
  "authors": [
    "Cole Graham"
  ],
  "comment": "38 pages",
  "categories": [
    "math.AP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-07T02:08:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35C07",
      "35K57"
    ],
    "keywords": [
      "fisher-kpp fronts",
      "precise asymptotics",
      "one-dimensional fisher-kpp equation",
      "long time",
      "step-like initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}