{
  "id": "1903.10756",
  "version": "v1",
  "published": "2019-03-26T09:38:36.000Z",
  "updated": "2019-03-26T09:38:36.000Z",
  "title": "Full family of flattening solitary waves for the mass critical generalized KdV equation",
  "authors": [
    "Yvan Martel",
    "Didier Pilod"
  ],
  "comment": "62 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "For the mass critical generalized KdV equation $\\partial_t u + \\partial_x (\\partial_x^2 u + u^5)=0$ on $\\mathbb R$, we construct a full family of flattening solitary wave solutions. Let $Q$ be the unique even positive solution of $Q''+Q^5=Q$. For any $\\nu\\in (0,\\frac 13)$, there exist global (for $t\\geq 0$) solutions of the equation with the asymptotic behavior \\begin{equation*} u(t,x)= t^{-\\frac{\\nu}2} Q\\left(t^{-\\nu} (x-x(t))\\right)+w(t,x) \\end{equation*} where, for some $c>0$, \\begin{equation*} x(t)\\sim c t^{1-2\\nu} \\quad \\mbox{and}\\quad \\|w(t)\\|_{H^1(x>\\frac 12 x(t))} \\to 0\\quad \\mbox{as $t\\to +\\infty$.} \\end{equation*} Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-26T09:38:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q53",
      "35B40",
      "37K40"
    ],
    "keywords": [
      "mass critical generalized kdv equation",
      "full family",
      "initial data",
      "flattening solitary wave solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}