Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities

Abstract

In a 1979 paper, Okamoto introduced the space of initial values for the six Painlevé 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-Painlevé property), which is a generalisation of the Painlevé 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.

Regularising transformations for the $$(n,\,n+1)$$-Liénard equations

“…We start from the system q = p, p = 2 j=0 f j q j p + 3 j=0 g j q j . (8) In this case the finite cascade is longer. We find…”

“…We discuss two ways to regularise the system, one by interchanging the dependent and independent variables and the other one using the hodograph transformation. We conjecture that the last transformation can also be used in a wider classes of equations, for instance in equatons with solutions possessing movable algebraic singularities as in [8].…”

“…The geometric approach for the Painlevé equations was developed in the papers [9,10] (see also [7]). The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations was also applied to obtain information about the singularity structure of solutions of the corresponding non-linear differential equations in [5,8]. Originally, the notion of a blow-up comes from algebraic geometry.…”

“…When the cascade is finite, we examine the resulting systems. If they are not regular on the last exceptional line, one may find a way to regularise them (for instance, by interchanging the dependent and independent variables as in [8] or examining the n-th order regularisability in case of algebroid solutions [6]) and extract information about the nature of singularities of the original system (e.g., solutions have movable algebraic branch points either locally or globally). Keeping track of the changes of variables introduced by the blow-ups in a given finite cascade one then finds a bi-rational regularising transformation for the system under consideration.…”

“…For the Painlevé equations the process of repeatedly blowing up base points is finite and leads to the construction of regularising transformations [9]. For certain classes of equations with the quasi-Painlevé property this process is also finite but the regularisation involves the interchanging of dependent and independent variables [8]. A natural question is how to extend such results to other classes of equations.…”