Degeneration scheme of 4-dimensional Painlev\'e-type equations

TL;DR

This paper systematically derives 4-dimensional Painlevé-type equations through degeneration of known equations, classifying 22 types and including well-known systems like Noumi-Yamada, with Hamiltonian formulations.

Contribution

It provides a comprehensive degeneration scheme for 4D Painlevé-type equations, expanding the classification and connecting them via Hamiltonian structures.

Findings

Identified 22 types of 4D Painlevé-type equations.

Included well-known systems such as Noumi-Yamada.

Presented Hamiltonian forms using classical Painlevé Hamiltonians.

Abstract

Four 4-dimensional Painlev\'e-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 Painlev\'e system. Degenerating these four source equations, we systematically obtained other 4-dimensional Painlev\'e-type equations. If we only consider Painlev\'e-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painlev\'e-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 Painlev\'e equations.