{
  "id": "2407.02717",
  "version": "v1",
  "published": "2024-07-02T23:53:46.000Z",
  "updated": "2024-07-02T23:53:46.000Z",
  "title": "A Direct Construction of Solitary Waves for a Fractional Korteweg-de Vries Equation With an Inhomogeneous Symbol",
  "authors": [
    "Swati Yadav",
    "Jun Xue"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct solitary waves for the fractional Korteweg-De Vries type equation $u_t + (\\Lambda^{-s}u + u^2)_x = 0$, where $\\Lambda^{-s}$ denotes the Bessel potential operator $(1 + |D|^2)^{-\\frac{s}{2}}$ for $s \\in (0,\\infty)$. The approach is to parameterise the known periodic solution curves through the relative wave height. Using a priori estimates, we show that the periodic waves locally uniformly converge to waves with negative tails, which are transformed to the desired branch of solutions. The obtained branch reaches a highest wave, the behavior of which varies with $s$. The work is a generalisation of recent work by Ehrnstr\\\"om-Nik-Walker, and is as far as we know the first simultaneous construction of small, intermediate and highest solitary waves for the complete family of (inhomogeneous) fractional KdV equations with negative-order dispersive operators. The obtained waves display exponential decay rate as $|x| \\to \\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-02T23:53:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76B25",
      "35C07",
      "76B03"
    ],
    "keywords": [
      "fractional korteweg-de vries equation",
      "solitary waves",
      "direct construction",
      "display exponential decay rate",
      "inhomogeneous symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}