The isomonodromic deformation equations such as the Painlevé 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
Painlevé test/Painlevé property, reduction of integrable hierarchy, the
Lax formulation, algebro-geometric methods, and others. Among them, the
Padé 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 Padé 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 Padé 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 Padé 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.
