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.
Highlights
Differential equations and their solutions in the complex plane have been studied extensively since the 19th century
Briot and Bouquet [1] noted that cases where a differential equation can be integrated directly are extremely rare, and one should study the properties of the solutions of a differential equation through the equation itself, as they have demonstrated for elliptic functions
The case is more involved for non-linear differential equations, for which singularities can develop somewhat spontaneously, depending on the initial data, and a priori the nature of the singularities cannot be determined by inspecting the differential equation
Summary
In the papers cited above, Filipuk and Halburd prove that the conditions, within the given classes of equations, are sufficient for all movable singularities to be algebraic poles of the form (2), with the proviso that these are reachable by analytic continuation along a path of finite length. The study of this property was continued by one of the authors for other classes of second-order equations [17] and certain Hamiltonian systems [19]. Employing a method originating in algebraic geometry, called a blow-up, we will resolve certain points of indeterminacy, or base points, which an equivalent system of equations acquires in an augmented (compact) phase space which includes the points at infinity in the space of dependent
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