Elements of Higher-Order Linear Equations

2014 ◽  
pp. 171-256
Keyword(s):  
Linear Equations ◽  
Higher Order ◽  
Order Linear

Generalized Halphen systems

2006 ◽  
Vol 136 (6) ◽  
pp. 1287-1301
Author(s):  
Nalini Joshi ◽  
Andrew Pickering
Keyword(s):  
Nineteenth Century ◽  
Nonlinear Equations ◽  
Linear Equations ◽  
Higher Order ◽  
Systems Of Nonlinear Equations ◽  
Weierstrass Function ◽  
Compatibility Conditions ◽  
Order Linear ◽  
Painlevé Transcendent ◽  
Painlevé Property

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.

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