Monodromy- and spectrum-preserving deformations I

Research output: Contribution to journalArticle

313 Scopus citations

Abstract

A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painlevé equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painlevé equation family is shown to be -d/dx ln(Δ+-), where Δ+ and Δ- are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system. The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations; additional results will be presented in a sequence of future papers.

Original languageEnglish (US)
Pages (from-to)65-116
Number of pages52
JournalCommunications in Mathematical Physics
Volume76
Issue number1
DOIs
StatePublished - Feb 1 1980

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Fingerprint Dive into the research topics of 'Monodromy- and spectrum-preserving deformations I'. Together they form a unique fingerprint.

  • Cite this