{
  "id": "2103.02897",
  "version": "v1",
  "published": "2021-03-04T08:53:51.000Z",
  "updated": "2021-03-04T08:53:51.000Z",
  "title": "Stability of traveling waves for the Burgers-Hilbert equation",
  "authors": [
    "Ángel Castro",
    "Diego Córdoba",
    "Fan Zheng"
  ],
  "comment": "57 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider smooth solutions of the Burgers-Hilbert equation that are a small perturbation $\\delta$ from a global periodic traveling wave with small amplitude $\\epsilon$. We use a modified energy method to prove the existence time of smooth solutions on a time scale of $\\frac{1}{\\epsilon\\delta}$ with $0<\\delta\\ll\\epsilon\\ll1$ and on a time scale of $\\frac{\\epsilon}{\\delta^2}$ with $0<\\delta\\ll\\epsilon^2\\ll1$. Moreover, we show that the traveling wave exists for an amplitude $\\epsilon$ in the range $(0,\\epsilon^*)$ with $\\epsilon^*\\sim 0.29$ and fails to exist for $\\epsilon>\\frac{2}{e}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-04T08:53:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "burgers-hilbert equation",
      "smooth solutions",
      "time scale",
      "global periodic traveling wave",
      "existence time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}