A local asymptotic analysis of the discrete first Painlevé equation as the discrete independent variable approaches infinity (Q1381210)

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.