Nagao H., Yamada Y. Padé Methods for Painlevé Equations

Springer, 2021. — 94 p.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.Preface
Padé Approximation and Differential Equation
Linear Differential Equations
A Toy Example of Padé Approximation
Contiguity Relations
Explicit Solutions
Padé Approximation for PVI
Derivation of the Differential Equation
Deformation Equation
Explicit Solutions by Schur Functions
Extension to Garnier System
More on the Schur Functions
Appendix
Padé Approximation for q-Painlevé/Garnier Equations
Lax Pair for the q-Garnier Equation
The L1 Equation
Special Solutions
Relation to 2 times2 Lax Form
Padé Interpolation
Cauchy–Jacobi Formula
Application to q-Garnier System
Special Solutions
A Duality Between q-Appell–Lauricella and q-HG Series
Padé Interpolation on q-Quadratic Grid
Contiguous Relations
Lax Pair and the Compatibility
Special Solutions
Multicomponent Generalizations
Padé Approximations With Multicomponent
Application to the N-Garnier System
Discrete Mahler Duality
Appendix References
Index