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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.
EXPONENTIAL ASYMPTOTICS AND CAPILLARY WAVES ∗
"... Abstract. Recently developed techniquesin exponential asymptoticsbeyond all ordersare employed on the problem of potential flows with a free surface and small surface tension, in the absence of gravity. Exponentially small capillary wavesare found to be generated on the free surface where the equipo ..."
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Abstract. Recently developed techniquesin exponential asymptoticsbeyond all ordersare employed on the problem of potential flows with a free surface and small surface tension, in the absence of gravity. Exponentially small capillary wavesare found to be generated on the free surface where the equipotentialsfrom singularitiesin the flow (for example, stagnation pointsand corners) meet it. The amplitude of these wavesisdetermined, and the implicationsare considered for many quite general flows. The asymptotic results are compared to numerical simulations of the full problem for flow over a polygonal plough and for flow round a rightangled corner, and they show remarkably good agreement, even for quite large values of the surface tension parameter.
Acta Applicandae Mathematicae 56: 1–98, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. The Devil’s Invention: Asymptotic, Superasymptotic and Hyperasymptotic
, 1998
"... Abstract. Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, an ..."
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Abstract. Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a ‘hyperasymptotic ’ approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescaleandadd process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration. Mathematics Subject Classifications (1991): 34E05, 40G99, 41A60, 65B10. Key words: perturbation methods, asymptotic, hyperasymptotic, exponential smallness. “Divergent series are the invention of the devil, and it is shameful to base on