{
  "id": "0912.4867",
  "version": "v2",
  "published": "2009-12-24T14:17:15.000Z",
  "updated": "2010-09-07T14:20:09.000Z",
  "title": "hbar-expansion of KP hierarchy: Recursive construction of solutions",
  "authors": [
    "Kanehisa Takasaki",
    "Takashi Takebe"
  ],
  "comment": "29 pages; Minor changes",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "math.QA",
    "nlin.SI"
  ],
  "abstract": "The \\hbar-dependent KP hierarchy is a formulation of the KP hierarchy that depends on the Planck constant \\hbar and reduces to the dispersionless KP hierarchy as \\hbar -> 0. A recursive construction of its solutions on the basis of a Riemann-Hilbert problem for the pair (L,M) of Lax and Orlov-Schulman operators is presented. The Riemann-Hilbert problem is converted to a set of recursion relations for the coefficients X_n of an \\hbar-expansion of the operator X = X_0 + \\hbar X_1 + \\hbar^2 X_2 +... for which the dressing operator W is expressed in the exponential form W = \\exp(X/\\hbar). Given the lowest order term X_0, one can solve the recursion relations to obtain the higher order terms. The wave function \\Psi associated with W turns out to have the WKB form \\Psi = \\exp(S/\\hbar), and the coefficients S_n of the \\hbar-expansion S = S_0 + \\hbar S_1 + \\hbar^2 S_2 +..., too, are determined by a set of recursion relations. This WKB form is used to show that the associated tau function has an \\hbar-expansion of the form \\log\\tau = \\hbar^{-2}F_0 + \\hbar^{-1}F_1 + F_2 + ...",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-09-07T14:20:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37K10"
    ],
    "keywords": [
      "kp hierarchy",
      "recursive construction",
      "recursion relations",
      "hbar-expansion",
      "wkb form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 841173,
      "adsabs": "2009arXiv0912.4867T"
    }
  }
}