{
  "id": "1106.1991",
  "version": "v2",
  "published": "2011-06-10T09:30:22.000Z",
  "updated": "2011-09-29T19:12:27.000Z",
  "title": "The space of 4-ended solutions to the Allen-Cahn equation on the plane",
  "authors": [
    "Michal Kowalczyk",
    "Yong Liu",
    "Frank Pacard"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "An entire solution of the Allen-Cahn equation $\\Delta u=F'(u)$, where $F$ is an even, bistable function, is called a $2k$-end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks like the one dimensional, heteroclinic solution. In this paper we initiate a program to classify the four-end solutions of the Allen-Cahn equation in $\\R^2$. We show that there exists a one parameter family of solutions containing the saddle solution, for which the angle between the nodal lines is $\\frac{\\pi}{2}$, as well as solutions for which the angle between the asymptotic half lines is any $\\theta\\in (0, \\frac{\\pi}{2})$. This justifies the definition of the angle map for a four-end solution $u$, which is the angle $\\theta=\\theta(u)\\in (0, \\frac{\\pi}{2})$ between the asymptote to the nodal line in the first quadrant and the x axis. Then we show that on any connected component in the moduli space of four-end solutions the angle map is surjective onto $(0,\\frac{\\pi}{2})$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-09-29T19:12:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "allen-cahn equation",
      "four-end solution",
      "angle map",
      "nodal line",
      "asymptotic half lines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.1991K"
    }
  }
}