{
  "id": "1009.4042",
  "version": "v2",
  "published": "2010-09-21T10:15:31.000Z",
  "updated": "2015-03-23T16:25:50.000Z",
  "title": "Uniqueness and Nondegeneracy of Ground States for $(-Δ)^s Q + Q - Q^{α+1} = 0$ in $\\mathbb{R}$",
  "authors": [
    "Rupert L. Frank",
    "Enno Lenzmann"
  ],
  "comment": "45 pages",
  "journal": "Acta Math. 210 (2013), no. 2, 261-318",
  "doi": "10.1007/s11511-013-0095-9",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove uniqueness of ground state solutions $Q = Q(|x|) \\geq 0$ for the nonlinear equation $(-\\Delta)^s Q + Q - Q^{\\alpha+1}= 0$ in $\\mathbb{R}$, where $0 < s < 1$ and $0 < \\alpha < \\frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \\alpha < \\infty$ for $s \\geq 1/2$. Here $(-\\Delta)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $\\alpha=1$ in [Acta Math., \\textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\\Delta)^s + 1 - (\\alpha+1) Q^\\alpha$ is nondegenerate; i.\\,e., its kernel satisfies $\\mathrm{ker}\\, L_+ = \\mathrm{span}\\, \\{Q'\\}$. This result about $L_+$ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-21T10:15:31.000Z",
      "title": "Uniqueness and Nondegeneracy of Ground States for $(-Δ)^s Q + Q - Q^{α+1} = 0$ in $\\R$",
      "abstract": "We prove uniqueness of ground state solutions $Q = Q(|x|) \\geq 0$ for the nonlinear equation $(-\\Delta)^s Q + Q - Q^{\\alpha+1}= 0$ in $\\R$, where $0 < s < 1$ and $0 < \\alpha < \\frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \\alpha < \\infty$ for $s \\geq 1/2$. Here $(-\\Delta)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $\\alpha=1$ in [Acta Math., \\textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\\Delta)^s + 1 - (\\alpha+1) Q^\\alpha$ is nondegenerate; i.\\,e., its kernel satisfies $\\mathrm{ker}\\, L_+ = \\mathrm{span} \\, \\{ Q' \\}$. This result about $L_+$ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-23T16:25:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplacian",
      "nondegeneracy",
      "ground state solutions",
      "water wave equations",
      "specific uniqueness result"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.4042F"
    }
  }
}