A local asymptotic analysis of the discrete first Painlevé equation as the discrete independent variable approaches infinity
The discrete first Painlevé equation which appears in a model of quantum gravity is an integrable nonlinear nonautonomous difference equation which yields the well known first Painlevé equation in a continuum limit. The asymptotic study of its solutions as the discrete time-step \(n\to \infty\) is important both for physical applications and for checking the accuracy of its role as a numerical discretization of the Painlevé equation. In this paper it is shown that the asymptotic analysis carried out by \textit{P. Boutroux} [Ann. de l'Éc. Norm. (3) 30, 255-375 (1913; JFM 44.0382.02)] for the Painlevé equation as its independent variable approaches infinity also can be achieved for discrete Painlevé equation as its discrete independent variable approaches the same circuit.
integrable nonlinear nonautonomous difference equation
scientific article; zbMATH DE number 1129310