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 classification.

Key phrases

Bäcklund transformations. The Painlevé equations.

AMS Subject Classification (1991)

Primary: 34A34
Secondary: 34A20, 33E30


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