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The Painlevé equivalence problem for a constrained 3D system

Galina Filipuk, Michele Graffeo, Giorgio Gubbiotti, Alexander Stokes

Abstract

In this paper we propose a geometric approach to study Painlevé equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential equations arising in the theory of semi-classical orthogonal polynomials. We show that it can be restricted to a system of two first-order differential equations in two different ways on an invariant hypersurface. We build the space of initial conditions for each of these restricted systems and verify that they exhibit the Painlevé property from a geometric perspective. Utilising the Painlevé identification algorithm we also relate this system to the Painlevé VI equation and we build its global Hamiltonian structure. Finally, we prove that the autonomous limit of the original system is Liouville integrable, and the level curves of its first integrals are elliptic curves, which leads us to conjecture that the 3D system itself also possesses the Painlevé property without the need to restrict it to the invariant hypersurface.

The Painlevé equivalence problem for a constrained 3D system

Abstract

In this paper we propose a geometric approach to study Painlevé equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential equations arising in the theory of semi-classical orthogonal polynomials. We show that it can be restricted to a system of two first-order differential equations in two different ways on an invariant hypersurface. We build the space of initial conditions for each of these restricted systems and verify that they exhibit the Painlevé property from a geometric perspective. Utilising the Painlevé identification algorithm we also relate this system to the Painlevé VI equation and we build its global Hamiltonian structure. Finally, we prove that the autonomous limit of the original system is Liouville integrable, and the level curves of its first integrals are elliptic curves, which leads us to conjecture that the 3D system itself also possesses the Painlevé property without the need to restrict it to the invariant hypersurface.

Paper Structure

This paper contains 34 sections, 19 theorems, 160 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background material
  3. The 3D model
  4. Painlevé equations
  5. Intersection form and elementary transformations
  6. Spaces of initial conditions and the Painlevé property
  7. Generalised Halphen surfaces
  8. Hamiltonian structures of Painlevé equations
  9. The system on the hypersurface in the first parametrisation
  10. The first parametrisation and the associated 2D system
  11. Space of initial conditions
  12. Relation between the first 2D system and $\operatorname{P}_{\mathrm{VI}}$
  13. The system on the hypersurface in the second parametrisation
  14. The algebraic Darboux hypersurface
  15. Parametrisation of the hypersurface
  16. ...and 19 more sections

Key Result

Theorem I

The system syst3D, complemented with the invariant hypersurface condition eq:St restricts to a 2D system, admitting a space of initial conditions whose $(-2)$-curves intersect according to the Dynkin diagram $\mathop{\mathrm{D}}\nolimits_4^{(1)}$. In addition, we show:

Figures (9)

  • Figure 1: The Dynkin diagram of $\mathop{\mathrm{D}}\nolimits_4^{(1)}$.
  • Figure 2: The elementary transformation relating $\mathbb{F}_k$ and $\mathbb{F}_{k+1}$.
  • Figure 3: Sequence of blow ups and contraction from $(\mathbb{P}^1\times \mathbb{P}^1)_t$ to $X_t$. Curves of self-intersection $-1,-2,-3$ are coloured in red, blue, green respectively. See \ref{['rem:stricttransforms']} for notation.
  • Figure 4: Sequence of blow ups and surface for system \ref{['second_system']} with $\mathbb{F}_1$ compactification, with $(-1)$-curves in red and $(-2)$-curves in blue.
  • Figure 5: Coordinate change for $\mathbb{F}_1$ in order to obtain a local Hamiltonian structure for system \ref{['second_system']}, with locations of poles of $\omega_t$ in blue.
  • ...and 4 more figures

Theorems & Definitions (63)

  • Theorem I: \ref{['Th1', 'prop:darboux', 'cor:secondsystemtosystfg', 'prop:hamsys', 'prop:sys3d']}
  • Conjecture 1
  • Theorem 2.2: MinChen
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • Definition 2.7: Hirzebruch surface
  • Remark 2.8
  • ...and 53 more