Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations
Pilar R Gordoa, Nalini Joshi and Andrew Pickering
Research Report 98-28
Date: 2 September 1998
The truncation method is a collective name for techniques that arise
from truncating a Laurent series expansion (with leading term) of solutions
of nonlinear partial differential equations (PDEs). Despite its utility in
finding Bäcklund transformations and other remarkable properties
of integrable PDEs, it has not been generally extended
to ordinary differential equations (ODEs). Here we give a new universal
method that provides such an extension and show how to apply it to
the classical nonlinear ODEs called the Painlevé equations.
Our main new idea is to consider mappings that preserve the locations
of a natural subset of the movable poles admitted by the equation.
In this way we are able to recover all known fundamental Bäcklund
transformations for the equations considered. We are also able to
derive Bäcklund transformations onto other ODEs in the Painlevé
Bäcklund transformations. The Painlevé equations.
AMS Subject Classification (1991)
Secondary: 34A20, 33E30
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Sydney Mathematics and Statistics