{
  "id": "1607.08802",
  "version": "v1",
  "published": "2016-07-29T13:34:44.000Z",
  "updated": "2016-07-29T13:34:44.000Z",
  "title": "Refined long time asymptotics for Fisher-KPP fronts",
  "authors": [
    "James Nolen",
    "Jean-Michel Roquejoffre",
    "Lenya Ryzhik"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the one-dimensional Fisher-KPP equation, with an initial condition $u_0(x)$ that coincides with the step function except on a compact set. A well-known result of M. Bramson states that, as $t\\to+\\infty$, the solution converges to a traveling wave located at the position $X(t)=2t-(3/2)\\log t+x_0+o(1)$, with the shift $x_0$ that depends on $u_0$. U. Ebert and W. Van Saarloos have formally derived a correction to the Bramson shift, arguing that $X(t)=2t-(3/2)\\log t+x_0-3\\sqrt{\\pi}/\\sqrt{t}+O(1/t)$. Here, we prove that this result does hold, with an error term of the size $O(1/t^{1-\\gamma})$, for any $\\gamma>0$. The interesting aspect of this asymptotics is that the coefficient in front of the $1/\\sqrt{t}$-term does not depend on $u_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-29T13:34:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35C07",
      "35B40"
    ],
    "keywords": [
      "refined long time asymptotics",
      "fisher-kpp fronts",
      "one-dimensional fisher-kpp equation",
      "bramson states",
      "bramson shift"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}