Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type

TL;DR

This work develops a geometric framework to identify Hamiltonian systems of quasi-Painlevé type by constructing Okamoto-like spaces of initial conditions via cascades of blow-ups on $\mathbb{CP}^2$ and comparing the irreducible components of inaccessible divisors. By matching these divisor configurations, the authors derive explicit bi-rational transformations between distinct Hamiltonian representations that share the same global Hamiltonian structure, with applications to quasi-Painlevé II and IV types and systems exhibiting square-root and ordinary poles, including mixed singularities. An auxiliary function $W$ is introduced to certify the quasi-Painlevé property by proving certain intermediate divisors are inaccessible, yielding necessary coefficient constraints for algebraic movable singularities. The results extend the Okamoto-Sakai geometric paradigm from Painlevé equations to quasi-Painlevé systems and provide concrete links between multiple Hamiltonian realizations, illustrating a path toward a systematic classification of such systems.

Abstract

Some new Hamiltonian systems of quasi-Painlevé type are presented and the analogue of Okamoto's space of initial conditions computed. Using the geometric approach that was introduced originally for the identification problem of Painlevé equations, comparing the irreducible components of the inaccessible divisors arising in the blow-up process, we find bi-rational coordinate changes between some of these systems that give rise to the same global Hamiltonian structure. This scheme thus gives a method for identifying Hamiltonian systems up to bi-rational maps, which is performed in this article for systems of quasi-Painlevé type having singularities that are either square-root type algebraic poles or ordinary poles.

Figures (11)

• Figure 1: Geometry of the curve portrait associated with the equivalent systems in \ref{['eq:nonind_zero']} (left) and in \ref{['eq:ind_zero']} (right). We mark with green crosses the equilibrium points along the lines $x=0$ and $y=0$ represented in magenta.
• Figure 2: With the blow-up transformation the point $p$ is replaced by the line in blue, a curve $\mathbb{CP}^1$. All the directions of the flow lines converging at the point $p$ are represented by distinct points on the blue curve after the transformation.
• Figure 3: Blow-up cascades for $H_1^{\text{P}_{\text{I\!I}}}$ and space of initial conditions. On the right, in green the curves with self-intersection $-2$, in blue with self-intersection $-1$, in gray with self-intersection $\ge 0$.
• Figure 4: Blow-up cascades for $H_2^{\text{P}_{\text{I\!I}}}$ and space of initial conditions. On the right, in green the curves with self-intersection $-2$, in blue with self-intersection $-1$, in gray with self-intersection $\ge 0$.
• Figure 5: Blow-up cascades for $H_3^{\text{P}_{\text{I\!I}}}$ and $H_4^{\text{P}_{\text{I\!I}}}$ and relative space of initial conditions. On the right, in magenta the curves with self-intersection $-3$, in green with self-intersection $-2$, in blue with self-intersection $-1$, in gray with self-intersection $\ge 0$.

Theorems & Definitions (1)

• Lemma 2.1