Title: Nonlinear $q$-Stokes phenomena for $q$-Painlevé I Show Chinese title

Title: 非线性$q$-Stokes 现象对于$q$-Painlevé I

Abstract: We consider the asymptotic behaviour of solutions of the first $q$-difference Painlev\'{e} equation in the limits $|q|\rightarrow 1$ and $n\rightarrow\infty$. Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the Stokes curves are described by curves known as $q$-spirals. Show Chinese abstract

Abstract: 我们考虑第一 $q$-差分 Painlevé 方程在极限 $|q|\rightarrow 1$和 $n\rightarrow\infty$下的渐近行为。 使用渐近幂级数，我们描述了四个包含隐藏的超越阶自由参数的解族。 这些渐近解表现出斯托克斯现象，这通常在经典幂级数方法中是不可见的。 为了研究这种现象，我们应用指数渐近技术来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，我们还确定了复平面上渐近行为由幂级数表达式描述的区域，并发现斯托克斯曲线是由称为 $q$-螺旋线的曲线描述的。