Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities

TL;DR

This paper extends the concept of initial value spaces to certain complex differential equations with algebraic poles, providing a method to identify equations without movable logarithmic singularities.

Contribution

It generalizes Okamoto's initial value space construction to equations with algebraic poles, offering an algorithm to detect equations free of movable logarithmic branch points.

Findings

Constructs regular initial value problems after variable transformations.

Identifies equations with the quasi-Painlevé property.

Provides an algorithm for detecting movable logarithmic singularities.

Abstract

In a 1979 paper, K. Okamoto introduced the space of initial values for the six Painlev\'e equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase space, a rational surface with certain exceptional divisors removed. We show that the construction of the space of initial values remains meaningful for certain classes of second-order complex differential equations, and more generally, Hamiltonian systems, where all movable singularities of all their solutions are algebraic poles (by some authors denoted the quasi-Painlev\'e property), which is a generalisation of the Painlev\'e property. The difference here is that the initial value problems obtained in the extended phase space become regular only after an additional change of dependent and independent variables. Constructing the analogue of space of initial values for these equations in this way also serves as an algorithm to single out, from a given class of equations or system of equations, those equations which are free from movable logarithmic branch points.