{
  "id": "2406.15148",
  "version": "v1",
  "published": "2024-06-21T13:46:27.000Z",
  "updated": "2024-06-21T13:46:27.000Z",
  "title": "Remarks on solitary waves in equations with nonlocal cubic terms",
  "authors": [
    "Johanna Ulvedal Marstrander"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form \\begin{equation*} \\partial_t u + \\partial_x(\\Lambda^s u + u\\Lambda^r u^2) = 0, \\end{equation*} where $\\Lambda^s, \\Lambda^r$ are Bessel-type Fourier multipliers. The linear operator may be of low fractional order, $s>0$, while the operator on the nonlinear part is assumed to act slightly smoother, $r<s-1$. The problem is related to the mathematical theory of water waves; we build upon previous works on similar equations, extending them to allow for a nonlocal nonlinearity. Mathematical tools include constrained minimization, Lion's concentration-compactness principle, spectral estimates, and product estimates in fractional Sobolev spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-21T13:46:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76B15",
      "76B25",
      "35A15"
    ],
    "keywords": [
      "nonlocal cubic terms",
      "solitary waves",
      "fractional sobolev spaces",
      "smooth solitary-wave solutions",
      "lions concentration-compactness principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}