Pade Methods for Painleve Equations

The isomonodromic deformation equations such as the Painleve and Garnier systems are an important class of nonlinear differential equations in mathematics and mathematical physics. For discrete analogs of these equations in particular, much progress has been made in recent decades. Various approaches to such isomonodromic equations are known: the Painleve test/Painleve property, reduction of integrable hierarchy, the Lax formulation, algebro-geometric methods, and others. Among them, the Pade method explained in this book provides a simple approach to those equations in both continuous and discrete cases.For a given function f(x), the Pade approximation/interpolation supplies the rational functions P(x), Q(x) as approximants such as f(x)~P(x)/Q(x). The basic idea of the Pade method is to consider the linear differential (or difference) equations satisfied by P(x) and f(x)Q(x). In choosing the suitable approximation problem, the linear differential equations give the Lax pair for some isomonodromic equations. Although this relation between the isomonodromic equations and Pade approximations has been known classically, a systematic study including discrete cases has been conducted only recently. By this simple and easy procedure, one can simultaneously obtain various results such as the nonlinear evolution equation, its Lax pair, and their special solutions. In this way, the method is a convenient means of approaching the isomonodromic deformation equations.