Pade Methods for Painleve Equations

The Padé method is a systematic approach to isomonodromic deformation equations, providing rational functions as approximants for given functions. It offers a way to obtain nonlinear evolution equations, their Lax pairs, and special solutions by considering linear differential equations satisfied by the approximants. This method simplifies the study of isomonodromic equations and their applications in mathematics and mathematical physics.

Format: Paperback / softback
Length: 90 pages
Publication date: 02 September 2021
Publisher: Springer Verlag, Singapore

The isomonodromic deformation equations, including the Painlevé and Garnier systems, are a significant category of nonlinear differential equations in mathematics and mathematical physics. In recent years, remarkable advancements have been made in the study of discrete analogs of these equations. Various approaches to such isomonodromic equations have been developed, such as the Painlevé test/Painlevé property, reduction of integrable hierarchy, the Lax formulation, algebro-geometric methods, and others. Among these, the Padé method presented in this book offers a straightforward approach to both continuous and discrete cases of isomonodromic equations.

For a given function f(x), the Padé approximation/interpolation provides rational functions P(x) and Q(x) as approximants, such as f(x)~P(x)/Q(x). The fundamental concept of the Padé method involves considering the linear differential (or difference) equations satisfied by P(x) and f(x)Q(x). By selecting the appropriate approximation problem, the linear differential equations yield the Lax pair for certain isomonodromic equations. While this relation between isomonodromic equations and Padé approximations has been recognized in the classical literature, a systematic investigation, including discrete cases, has only recently been conducted. Through this simple and efficient procedure, various results can be obtained simultaneously, including the nonlinear evolution equation, its Lax pair, and their special solutions.

In this manner, the Padé method serves as a convenient tool for approaching the isomonodromic deformation equations. Its simplicity and effectiveness make it a valuable resource for researchers and practitioners in the field.

Weight: 168g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811629976
Edition number: 1st ed. 2021