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Generalized Halphen systems

Published online by Cambridge University Press:  12 July 2007

Nalini Joshi
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW2006 Sydney, Australia ([email protected])
Andrew Pickering
Affiliation:
Departamento de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, C/Tulipán s/n, 28933 Móstoles-Madrid, Spain ([email protected])

Abstract

Towards the end of the nineteenth century, Halphen studied a remarkable sequence of higher-order linear equations with doubly periodic coefficients, generalizations of a certain Lamé equation, having the property that quotients of solutions are single valued. Here we consider further generalizations where, instead of the Weierstrass ℘-function, the coefficients depend on the first Painlevé transcendent. Using these equations, we obtain new higher-order systems of nonlinear equations having the Painlevé property. We also give new results on the interpretation of the Painlevé tests with regard to the representations of solutions, general and particular, afforded by various branches, and to understanding the corresponding pattern of compatibility conditions.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2006

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