{
  "id": "2312.04130",
  "version": "v1",
  "published": "2023-12-07T08:37:47.000Z",
  "updated": "2023-12-07T08:37:47.000Z",
  "title": "The Wave Equation on Lattices and Oscillatory Integrals",
  "authors": [
    "Cheng Bi",
    "Jiawei Cheng",
    "Bobo Hua"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we establish sharp dispersive estimates for the linear wave equation on the lattice $\\mathbb{Z}^d$ with dimension $d=4$. Combining the singularity theory with results in uniform estimates of oscillatory integrals, we prove that the optimal time decay rate of the fundamental solution is of order $|t|^{-\\frac{3}{2}}\\log |t|$, which is the first extension of P. Schultz's results \\cite{S98} in $d=2,3$ to the higher dimension. We also observe that the decay rate for $d=2,3,4$ can be well interpreted by the Newton polyhedra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-07T08:37:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "oscillatory integrals",
      "optimal time decay rate",
      "linear wave equation",
      "singularity theory",
      "newton polyhedra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}