Isomonodromic Deformations and Painlevé Equations

Abstract

The connection between isomonodromic deformation of the Fuchsian system of linear differential equations and the Painleve VI equation is considered. Namely, any Fuchsian system can be transformed in accordance with some dynamical system that is called the Schlesinger equations. Such deformation can be considered as a flow on the space of Fuchsian systems; it is the Schlesinger flow. It is shown that the space of all Fuchsian equations can be rationally projected on the symplectic space \(\mathbb {C}\times \mathbb {C}\) in such a way that the preimage of any point consists of the systems with the same monodromy. The Schlesinger flow can be projected on this symplectic space because the corresponding vector field has the following property: the projections of the values of the field at points coincide if the projections of the points coincide. The resulting flow on the extended phase space \(\mathbb {C}\times \mathbb {C}\times \mathbb {C}\ni (p,q,t)\) is called a Painleve VI Hamiltonian system. The nonlinear ODE Painleve VI is the Euler–Lagrange equation corresponding to the Hamiltonian equations under the Legendre transformation. The compatibility of the Schlesinger deformation with the initial Fuchsian system proves that the Painleve VI implies the constant monodromy; the converse does not hold because of the counterexample. The foregoing theory can be considered as an explanation to the presenting of the system of two compatible linear systems and a verification of the fact that the compatibility condition is Painleve VI. Other various linear systems with four poles, taking into account their orders, are also considered, and the corresponding pairs of the compatible linear systems are presented. It is verified directly that the compatibility conditions give all the other equations from Painleve’s list.