{
  "id": "2112.11278",
  "version": "v3",
  "published": "2021-12-21T15:05:04.000Z",
  "updated": "2022-10-23T18:51:54.000Z",
  "title": "Asymptotic $N$-soliton-like solutions of the fractional Korteweg-de Vries equation",
  "authors": [
    "Arnaud Eychenne"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $$ \\partial_t u - \\partial_x\\left(|D|^{\\alpha}u - u^2 \\right)=0, $$ in the whole sub-critical range $\\alpha \\in]\\frac12,2[$. More precisely, if $Q_c$ denotes the ground state solution associated to fKdV evolving with velocity $c$, then given $0<c_1< \\cdots < c_N$, we prove the existence of a solution $U$ of (fKdV) satisfying $$ \\lim_{t\\to\\infty} \\| U(t,\\cdot) - \\sum_{j=1}^NQ_{c_j}(x-\\rho_j(t)) \\|_{H^{\\frac{\\alpha}2}}=0, $$ where $\\rho'_j(t) \\sim c_j$ as $t \\to +\\infty$. The proof adapts the construction of Martel in the generalized KdV setting [Amer. J. Math. 127 (2005), pp. 1103-1140]) to the fractional case. The main new difficulties are the polynomial decay of the ground state $Q_c$ and the use of local techniques (monotonicity properties for a portion of the mass and the energy) for a non-local equation. To bypass these difficulties, we use symmetric and non-symmetric weighted commutator estimates. The symmetric ones were proved by Kenig, Martel and Robbiano [Annales de l'IHP Analyse Non Lin\\'eaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-10-23T18:51:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q53",
      "35Q35",
      "35B40",
      "37K40"
    ],
    "keywords": [
      "fractional korteweg-de vries equation",
      "soliton-like solutions",
      "asymptotic",
      "lihp analyse non lineaire",
      "ground state solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}