Part 2. Degeneration scheme of 4-dimensional Painlevé-type equations

Abstract

This is a continuation of Part 1 by one of the authors. We have shown that there are four 4-dimensional Painlevé-type equations derived from isomonodromic deformation of the Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painlevé system. In Part 2, we degenerate these four source equations, and systematically obtain other 4-dimensional 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 the Noumi-Yamada systems are included in this list. They are expressed as Hamiltonian systems, and their Hamiltonians are simply written by using the Hamiltonians of the classical Painlevé equations.