{
  "id": "2301.13433",
  "version": "v1",
  "published": "2023-01-31T06:00:31.000Z",
  "updated": "2023-01-31T06:00:31.000Z",
  "title": "On well-posedness results for the cubic-quintic NLS on $\\mathbb{T}^3$",
  "authors": [
    "Yongming Luo",
    "Xueying Yu",
    "Haitian Yue",
    "Zehua Zhao"
  ],
  "comment": "11 pages. Comments are welcome!",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the periodic cubic-quintic nonlinear Schr\\\"odinger equation \\begin{align}\\label{cqnls_abstract} (i\\partial_t +\\Delta )u=\\mu_1 |u|^2 u+\\mu_2 |u|^4 u\\tag{CQNLS} \\end{align} on the three-dimensional torus $\\mathbb{T}^3$ with $\\mu_1,\\mu_2\\in \\mathbb{R} \\setminus\\{0\\}$. As a first result, we establish the small data well-posedness of \\eqref{cqnls_abstract} for arbitrarily given $\\mu_1$ and $\\mu_2$. By adapting the crucial perturbation arguments in \\cite{zhang2006cauchy} to the periodic setting, we also prove that \\eqref{cqnls_abstract} is always globally well-posed in $H^1(\\mathbb{T}^3)$ in the case $\\mu_2>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-31T06:00:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "well-posedness results",
      "cubic-quintic nls",
      "periodic cubic-quintic nonlinear",
      "small data well-posedness",
      "crucial perturbation arguments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}