Exponential asymptotics and Stokes lines in a partial differential equation
Exponential asymptotics and Stokes lines in a partial differential equation
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偏微分方程中的指数渐近和斯托克斯线
DOI：
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发表时间：
2005
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通讯作者：
David B Mortimer
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文献类型：
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作者：
B. V. J. Chapman;David B Mortimer
考虑了晶体生长几何模型所激发的奇摄动线性偏微分方程。傅里叶变换解的最陡下降分析确定了鞍点、端点和极点的渐近贡献，以及这些点可以被打开和关闭的斯托克斯线。这些结果直接从方程中导出，通过最佳地截断朴素扰动展开和平滑Stokes间断。分析揭示了两种新类型的斯托克斯开关：高阶斯托克斯线，这是一个斯托克斯线的渐近级数的后期项的近似，并开关或关闭斯托克斯线本身;和第二代斯托克斯线，其中一个次显性指数在主斯托克斯线打开本身是负责切换另一个较小的指数。Berk等人（Berk等人，1982，J.Math.Phys.23，988-1002）讨论的“新的”斯托克斯线是第二代斯托克斯线，而Aoki等人（Aoki等人，1998，In Microbocal analysis and complex Fourier analysis（ed.K.J.Math.Phys.23，988-1002）讨论的“消失的”斯托克斯线是第二代斯托克斯线。F. T. Kawai），pp. 165-176）被高阶斯托克斯线切断。
A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The ‘new’ Stokes lines discussed by Berk et al. (Berk et al. 1982 J. Math. Phys. 23, 988–1002) are second-generation Stokes lines, while the ‘vanishing’ Stokes lines discussed by Aoki et al. (Aoki et al. 1998 In Microlocal analysis and complex Fourier analysis (ed. K. F. T. Kawai), pp. 165–176) are switched off by a higher-order Stokes line.