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On the theory of Complex Rays
 SIAM Review
, 1997
"... The article surveys the application of complex ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredien ..."
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The article surveys the application of complex ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredient in this framework is the role played by Stokes' phenomenon in determining the regions of existence of complex rays. The identification of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterisations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex wkb expansion of these wavefields. Examples will be given for several cases of physical importance. 1 Introduction The objective of this paper is to lay down a systematic complexified theory of monochromatic highfrequency wave propagation. The computational and analytical usefulness of such ...
Exponential asymptotics and Stokes lines in a partial differential equation
, 2005
"... 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 whi ..."
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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 higherorder 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 secondgeneration 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. (1982) are secondgeneration Stokes lines, while the ‘vanishing ’ Stokes lines discussed by Aoki et al. (1998) are switched off by a higherorder Stokes line.