{
  "id": "math/9201261",
  "version": "v1",
  "published": "1992-01-01T00:00:00.000Z",
  "updated": "1992-01-01T00:00:00.000Z",
  "title": "A steepest descent method for oscillatory Riemann-Hilbert problems",
  "authors": [
    "Percy Deift",
    "Xin Zhou"
  ],
  "comment": "6 pages. Abstract added in migration.",
  "journal": "Bull. Amer. Math. Soc. (N.S.) 26 (1992) 119-124",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg de Vries (MKdV) equation, $$y_t-6y^2y_x+y_{xxx}=0,\\qquad -\\infty<x<\\infty,\\ t\\ge0, y(x,t=0)=y_0(x),$$ but it will be clear immediately to the reader with some experience in the field, that the method extends naturally and easily to the general class of wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schr\\\"odinger (NLS), and Boussinesq equations, etc., and also to ``integrable'' ordinary differential equations such as the Painlev\\'e transcendents.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1992-01-01T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "oscillatory riemann-hilbert problems",
      "steepest descent method",
      "inverse scattering method",
      "ordinary differential equations",
      "long-time behavior"
    ],
    "tags": [
      "journal article",
      "expository article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Bull. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1992math......1261D"
    }
  }
}