From continuous to discrete Painlevé equations. (English) Zbl 0794.34013
Summary: Discrete analogues of the Painlevé equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method. Here we analyze additional methods for deriving integrable non-autonomous discrete equations. These methods are based on the discrete AKNS approach, on the use of Bäcklund und Schlesinger transforms of the (continuous) Painlevé equations, and on discrete analogues of Miura transformations.
MSC:
34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
39A10 | Additive difference equations |