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

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.