TY - JOUR

T1 - Regularising transformations for complex differential equations with movable algebraic singularities

AU - Kecker, Thomas

AU - Filipuk, Galina

N1 - Funding Information:
GF acknowledges the support of the National Science Center (Poland) through the Grant OPUS 2017/25/B/BST1/00931. TK acknowledges support of the London Mathematical Society (LMS) and the Faculty of Mathematics, Informatics and Mechanics at the University of Warsaw (MIMUW) for travel Grants to visit Warsaw in the years 2014, 2015 and 2016; these visits, were this research was initiated, were essential for the success of the project.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/3/6

Y1 - 2022/3/6

N2 - In a 1979 paper, K. 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.

AB - In a 1979 paper, K. 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.

KW - space of initial values

KW - blow-up

KW - movable algebraic singularity

KW - complex differential equation

UR - http://www.scopus.com/inward/record.url?scp=85126131969&partnerID=8YFLogxK

U2 - 10.1007/s11040-022-09417-6

DO - 10.1007/s11040-022-09417-6

M3 - Article

AN - SCOPUS:85126131969

SN - 1385-0172

VL - 25

JO - Mathematical Physics, Analysis and Geometry

JF - Mathematical Physics, Analysis and Geometry

IS - 1

M1 - 9

ER -