Introduction to the Theory of Isomonodromic Deformations of Linear Ordinary Differential Equations with Rational Coefficients

Abstract

The general solution of a linear ordinary differential equation (ODE) with rational coefficients is generically multivalued. This property is described by a representation of the fundamental homotopy group of the complex plane deprived of the singular points, the monodromy representation. The idea of isomonodromic deformations, which traces back to Riemann in the middle of the nineteenth century, is to construct a family of linear ODEs that share a given monodromy representation. This leads to systems of linear partial differential equations, the integrability conditions of which are nonlinear differential equations that enjoy the Painlevé property. It is thus a powerful tool to associate linear with integrable nonlinear equations.

The aim of these lectures is to get a first insight into the problem and to provide explicit algorithms to solve it, starting with linear ODEs that possess regular as well as irregular singularities. The first part is devoted to Schlesinger’s theorem, which solves the regular case. As an application, the Lax pair of Pvi is constructed. The second part is devoted to the theorems of M. Jimbo, T. Miwa, and K. Ueno, dealing with irregular singularities. The application chosen in that case is the construction of a Lax pair for P_I. Finally, the direct isomonodromy method that exploits these results to solve connection problems is outlined.