Painlevé equations: from continuous to discrete
This nicely written paper surveys the development of the theory of continuous and discrete Painlevé equations analyzing their derivation, fundamental properties, and further extensions. In the introduction, the authors describe the revival of the interest in integrability of partial differential equations in the late sixties of the last century through the major advancements in the field and discovery of surprising connections between the reductions of integrable partial differential equations that could be written as second order ordinary differential equations and classes of equations studied by Painlevé and his school. They proceed by discussing the role played by singularities in the analysis of integrability of a given nonlinear ordinary differential equation. Then some important properties of continuous Painlevé equations are briefly reviewed. In the next section, the idea of discrete integrability is explained and four major approaches to derive discrete Painlevé equations are described. Analyzing the fundamental properties of such equations, the authors emphasize their striking similarity with the corresponding properties of continuous ones. The problem of a global geometrical description of discrete Painlevé equations is addressed along with the Sakai approach to their classification. In the final part of the paper, several extensions of Painlevé equations are mentioned. The paper concludes with an extensive list of references on the topic.
scientific article; zbMATH DE number 5563615