Generalisation of Bureau-Guillot systems with Painlevé transcendents in the coefficients
Marta Dell'Atti, Galina Filipuk
TL;DR
The paper generalises Bureau-Guillot systems by embedding Painlevé transcendents into the coefficient functions and leveraging Okamoto–Sakai regularisation to guarantee the Painlevé property. It develops a unifying, geometry-driven method based on Okamoto polynomial Hamiltonians and type I birational transformations to construct polynomial and rational first-order systems that are birationally equivalent to Painlevé Hamiltonians. The authors systematically build BG analogues for $(P_{ ext{I}})$–$(P_{ ext{VI}})$, including higher-degree cases, and demonstrate mixed and quasi-Painlevé extensions, with explicit cascades of blow-ups on the space of initial conditions. These results broaden the landscape of systems with the Painlevé property and provide a geometric framework for non-rational coefficients, potentially informing future work on discrete analogues and broader differential equations.
Abstract
We construct a generalisation of what we call Bureau-Guillot systems, i.e. systems of first order equations with coefficient functions being Painlevé transcendents. The same Painlevé equation is related to the system and it appears as regularising condition in the regularisation process. The systems considered are birationally equivalent to the Okamoto polynomial Hamiltonian systems with rational coefficients for Painlevé equations, hence they possess the Painlevé property. This work extends the results of Bureau-Guillot in a two-fold way. On one side, we consider polynomial systems with degree larger than 2 that are free of movable critical points. These systems contain not only transcendents $\text{P}_{\text{I}}$ and $\text{P}_{\text{II}}$ in the coefficients, but also transcendents $\text{P}_{\text{III}}$, $\text{P}_{\text{IV}}$, $\text{P}_{\text{V}}$ and $\text{P}_{\text{VI}}$ (and/or their derivatives). On the other side, we explore examples of rational systems with the Painlevé transcendents in the coefficients birationally equivalent to the Okamoto polynomial systems. Lastly, we present a simpler version of the change of variables to obtain the analogues of the Bureau-Guillot systems. In this framework we discuss generalisations including the mixed case: systems related to the equation $(\text{P}_{\text{J}})$, with ${\text{J}=\text{I}, \dots, \text{VI}}$, containing coefficient functions that are solutions to $(\text{P}_{\text{K}})$ with $\text{K}\neq \text{J}$. In the latter, the equation $(\text{P}_{\text{K}})$ appears as a regularising condition during the regularisation process. Although we are primarily interested in systems possessing the Painlevé property, we also briefly discuss an analogous construction for systems including coefficient functions solving quasi-Painlevé equation.