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Communications in Mathematical Physics

, Volume 76, Issue 1, pp 65–116 | Cite as

Monodromy- and spectrum-preserving deformations I

  • Hermann Flaschka
  • Alan C. Newell
Article

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.

Keywords

Neural Network Integral Equation Partial Differential Equation Ordinary Differential Equation Singular Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Hermann Flaschka
    • 1
  • Alan C. Newell
    • 1
  1. 1.Department of Mathematics and Computer ScienceClarkson College of TechnologyPotsdamUSA

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