On the relation between the continuous and discrete Painlevé equations
Painlevé and Gambier at the end of the nineteen century classified all second-order ordinary differential equations with no movable branch points in their general solution and found six families of new second-order differential equations with this property. They are called the Painlevé transcendents PI--PVI [see \textit{E. L. Ince}, Integration of ordinary differential equations (German). Mannheim: BI (1966; Zbl 0191.09801)]. These equations arise in a natural way in several unrelated fields and today they are the object of a very active research [see \textit{M. J. Ablowitz} and \textit{P. A. Clarkson}, Solitons, nonlinear evolution equations and inverse scattering. London Mathematical Society Lecture Note Series. 149. Cambridge *K) etc.: Cambridge University Press (1991; Zbl 0762.35001)]. The Painlevé transcendents PII--PVI depend on parameters and they admit Bäcklund transformations, i.e., some transformations of the parameters which send solutions to one of the equations to another. The paper under review is devoted to give an interpretation of some special families of Bäcklund transformations as discrete difference equations satisfied by the Painlevé transcendents. As is remarked by the authors, the method is analogous to the recurrence relations satisfied by the classical special functions (like Bessel or orthogonal polynomials). It should be very interesting to study the dependence of the transcendents with respect to a continuous parameter instead of a discrete one.
continuous and discrete Painlevé equations
scientific article; zbMATH DE number 1535321