Abstract
The most important non-linear ordinary differential equations are the following six Painlevé equations:
where α, β, γ, δ are complex constants. (Warning: The parameters of P VI are slitely different from those customarily used; -β and ½ - δ have been denoted by β and δ. The reason of our choice will turn out to be clear in the text.) We study, in this chapter, these differential equations first classically (§1) and secondly in the framework of the monodromy preserving deformation. After introducing the concept of monodromy preserving deformation (§2, §3), we derive the Gamier system written in the form of Hamiltonian system, which governs such deformation of a second order Fuchsian equation with n+3 singularities (§4).
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© 1991 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig
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Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M. (1991). Monodromy Preserving Deformation, Painlevé Equations and Garnier Systems. In: From Gauss to Painlevé. Aspects of Mathematics, vol 16. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-90163-7_3
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DOI: https://doi.org/10.1007/978-3-322-90163-7_3
Publisher Name: Vieweg+Teubner Verlag
Print ISBN: 978-3-322-90165-1
Online ISBN: 978-3-322-90163-7
eBook Packages: Springer Book Archive