Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from t runcating a Laurent series expansion (with leading term) of generic solutio ns of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of inte grable PDEs, it has not been generally extended to ordinary differential eq uations (ODEs). Here we give a new general method that provides such an ext ension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equ ation. In this way we are able to recover all known fundamental Backlund tr ansformations for the equations considered. We are also able to derive Back lund transformations onto other ODEs in the Painleve classification.