Joshi N, Lustri C J, Luu S
School of Mathematics and Statistics, F07 , The University of Sydney , New South Wales 2006, Australia.
Department of Mathematics , Macquarie University , Sydney, New South Wales, Australia.
Proc Math Phys Eng Sci. 2017 Feb;473(2198):20160539. doi: 10.1098/rspa.2016.0539. Epub 2017 Feb 22.
We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the and solutions of the second continuous Painlevé equation.
我们考虑当自变量趋于无穷大时，第二个离散Painlevé方程的渐近行为。利用渐近幂级数，我们找到了在复平面的某个区域内渐近无极点的解。这些渐近解呈现出斯托克斯现象，这在经典幂级数方法中通常是不可见的。随后，我们应用指数渐近技术来研究此类现象，并获得与斯托克斯曲线相关的快速切换行为的数学描述。通过这种分析，我们确定了复平面中渐近行为由幂级数表达式描述的区域，并发现这些渐近解的行为与第二个连续Painlevé方程的解具有许多共同特征。
