{
  "id": "2008.01509",
  "version": "v1",
  "published": "2020-08-04T13:22:30.000Z",
  "updated": "2020-08-04T13:22:30.000Z",
  "title": "From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies",
  "authors": [
    "Alexandre Krajenbrink"
  ],
  "comment": "41 pages",
  "categories": [
    "math-ph",
    "cond-mat.dis-nn",
    "cond-mat.stat-mech",
    "math.MP",
    "math.PR",
    "nlin.SI"
  ],
  "abstract": "As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This consists in a quasi-universal hierarchy of equations, partly unifying an integro-differential generalization of the Painlev\\'e II hierarchy, the finite-time solutions of the Kardar-Parisi-Zhang equation, multi-critical fermions at finite temperature and a notable solution to the Zakharov-Shabat system associated to the largest real eigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the explicit unique solution to the inverse scattering transform of the Zakharov-Shabat system in terms of a Fredholm determinant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T13:22:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fredholm determinant",
      "integro-differential hierarchies",
      "zakharov-shabat system",
      "explicit unique solution",
      "largest real eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}