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Physics © by Springer-Verlag 1980 Monodromy- and Spectrum-Preserving Deformations I
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@MISC{Flaschka_physics©,
author = {Hermann Flaschka and Alan C. Newell},
title = {Physics © by Springer-Verlag 1980 Monodromy- and Spectrum-Preserving Deformations I},
year = {}
}
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Abstract
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 Painleve 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 Painleve equation family is shown to be — — ln(A+/A__ \ 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.
Keyphrases
spectrum-preserving deformation rational solution broad class irregular singular point partial differential equation several class special case general expression recent work linear ordinary differential equation second painleve equation family painleve equation specific new result integrable hamiltonian system isospectral deformation partial differential equation additional result general initial value problem basic idea second type third type central idea future paper monodromy preserving deformation linear singular integral equation
