{
  "id": "1112.0601",
  "version": "v1",
  "published": "2011-12-02T22:58:29.000Z",
  "updated": "2011-12-02T22:58:29.000Z",
  "title": "An hbar-expansion of the Toda hierarchy: a recursive construction of solutions",
  "authors": [
    "Kanehisa Takasaki",
    "Takashi Takebe"
  ],
  "comment": "37 pages, no figures. arXiv admin note: substantial text overlap with arXiv:0912.4867",
  "journal": "Analysis and Mathematical Physics Volume 2, Number 2 (2012), 171-214",
  "doi": "10.1007/s13324-012-0026-5",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "math.QA",
    "nlin.SI"
  ],
  "abstract": "A construction of general solutions of the \\hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\\bar L,\\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and \\bar W. The dressing operators are set in an exponential form as W = e^{X/\\hbar} and \\bar W = e^{\\phi/\\hbar}e^{\\bar X/\\hbar}, and the auxiliary operators X,\\bar X and the function \\phi are assumed to have \\hbar-expansions X = X_0 + \\hbar X_1 + ..., \\bar X = \\bar X_0 + \\hbar\\bar X_1 + ... and \\phi = \\phi_0 + \\hbar\\phi_1 + .... The coefficients of these expansions turn out to satisfy a set of recursion relations. X,\\bar X and \\phi are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form \\Psi = e^{S/\\hbar} and \\bar\\Psi = e^{\\bar S/\\hbar}, which leads to an \\hbar-expansion of the logarithm of the tau function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-02T22:58:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37K10",
      "35Q53"
    ],
    "keywords": [
      "toda hierarchy",
      "recursive construction",
      "riemann-hilbert problem",
      "hbar-expansion",
      "dressing operators"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Analysis and Mathematical Physics",
      "year": 2012,
      "month": "Jun",
      "volume": 2,
      "pages": 171
    },
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1079956,
      "adsabs": "2012AnMP....2..171T"
    }
  }
}