论文标题
WKB方法的动态系统方法：简单的转折点
A dynamical systems approach to WKB-methods: The simple turning point
论文作者
论文摘要
在本文中，我们重新访问第二阶微分方程的经典线性转弯点问题$ε^2 x'' +μ（t）x = 0 $，使用$μ（0）= 0，\，\，μ'（0）\ ne 0 $ for $ 0 <$ 0 <ε\ ll 1 $。 $ t = 0 $编写为一阶系统，因此对应于连接双曲线和椭圆机制的转弯点。我们的主要结果是，我们为WBK提供了一种替代方法，该方法基于动态系统理论，包括GSPT和爆炸，我们桥接了慢速系统的双曲线和椭圆形理论。作为优势，我们只需要$μ$的有限平滑度。我们开发的方法将在双曲到涡流转折点的其他单数扰动问题中很有用。
In this paper, we revisit the classical linear turning point problem for the second order differential equation $ε^2 x'' +μ(t)x=0$ with $μ(0)=0,\,μ'(0)\ne 0$ for $0<ε\ll 1$. Written as a first order system, $t=0$ therefore corresponds to a turning point connecting hyperbolic and elliptic regimes. Our main result is that we provide an alternative approach to WBK that is based upon dynamical systems theory, including GSPT and blowup, and we bridge -- perhaps for the first time -- hyperbolic and elliptic theories of slow-fast systems. As an advantage, we only require finite smoothness of $μ$. The approach we develop will be useful in other singular perturbation problems with hyperbolic-to-elliptic turning points.
