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The discrete and continuous Painlevé VI hierarchy and the Garnier systems

Published online by Cambridge University Press:  19 July 2002

F. W. Nijhoff  and
A. J. Walker
F. W. Nijhoff
Affiliation:
Department of Applied Mathematics, The University of Leeds, Leeds LS2 9JT, UK e-mail: frank@amsta.leeds.ac.uk and amtajw@amsta.leeds.ac.uk
A. J. Walker
Affiliation:
Department of Applied Mathematics, The University of Leeds, Leeds LS2 9JT, UK e-mail: frank@amsta.leeds.ac.uk and amtajw@amsta.leeds.ac.uk
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Abstract

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We present a general scheme to derive higher-order members of the Painlevé VI (PVI) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 43 , Issue A , June 2001 , pp. 109 - 123
Copyright
© 2001 Glasgow Mathematical Journal Trust