Degeneration scheme of 4-dimensional Painlevé-type equations

Abstract

Four 4-dimensional Painlevé-type equations are obtained by isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painlevé system. Degenerating these four source equations, we systematically obtained other 4-dimensional Painlevé-type equations. If we only consider Painlevé-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painlevé-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equations. Some well-known equations such as Noumi-Yamada systems are included in this list. They are written as Hamiltonian systems, and their Hamiltonians are neatly written using Hamiltonians of the classical Painlevé equations.

References in corpus (5)

• Dual Isomonodromic Deformations and Moment Maps to Loop Algebras
• Irregular connections and Kac-Moody root systems
• Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $D_6^{(1)}$, II
• A Particular Solution of a Painlevé System in Terms of the Hypergeometric Function ${}_{n+1}{F}_n$
• Linear differential equations on ${\mathbb{P}}^1$ and root systems

Cited by in corpus (15)

• Quantum Curve and the First Painlevé Equation
• Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
• Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III: Garnier Systems and Fuji-Suzuki Systems
• Four-dimensional Painlevé-type equations associated with ramified linear equations I: Matrix Painlevé systems
• Four-dimensional Painlevé-type equations associated with ramified linear equations II: Sasano systems
• Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two
• "Quantizations" of isomonodromic Hamiltonian Garnier system with two degrees of freedom
• The Sigma Form for the PII Hierarchy
• Singular Values of Products of Ginibre Random Matrices
• Isomonodromic deformations: Confluence, Reduction $\&$ Quantisation
• Space of initial conditions for the four-dimensional Fuji-Suzuki-Tsuda system
• Weights, Kovalevskaya exponents and the Painlevé property
• Moduli of regular singular parabolic connections of spectral type on smooth projective curves
• Four-dimensional Painlevé-type difference equations
• Moduli spaces of meromorphic connections, quiver varieties, and integrable deformations