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11
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 of localised patterns and snaking bifurcation diagrams
 Phys. D
, 2009
"... Localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiplescales analysis beyond all orders. The model studied is the SwiftHohenberg equation of nonlinear optics, which is equivalent to the classical SwiftHohenberg equation with a quadratic and a c ..."
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Localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiplescales analysis beyond all orders. The model studied is the SwiftHohenberg equation of nonlinear optics, which is equivalent to the classical SwiftHohenberg equation with a quadratic and a cubic nonlinearity. Applying the asymptotic technique away from the Maxwell point first, it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sechtype amplitude. In the vicinity of the Maxwell point, the beyondallorders calculation yields the “pinning range ” of parameters where stable stationary fronts connect the homogeneous and periodic states. The full bifurcation diagram for localised patterns is then computed analytically, including snake and ladder bifurcation curves. This last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a downfront. To this end, a third, superslow spatial scale needs to be introduced, in which fronts appear as boundary layers. In addition, the location of the Maxwell point and the oscillation wave number of localised patterns are required to fourthorder accuracy in the oscillation amplitude. 1
Uniform Asymptotics for the Incomplete Gamma Functions Starting From Negative Values of the Parameters
"... We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the i ..."
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We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for fl(a; z) and \Gamma(a; z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section.
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.
Effects of Exponentially Small Terms in the Perturbation Approach to Localized Buckling
, 1999
"... this paper is organized as follows. We start with the perturbation results of Wadee et al. (1997) and point out the flaw which renders the approach unable to account for all but the primary localized solutions. In x3 we detail how the procedure must be amended in the light of this finding and, cruci ..."
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this paper is organized as follows. We start with the perturbation results of Wadee et al. (1997) and point out the flaw which renders the approach unable to account for all but the primary localized solutions. In x3 we detail how the procedure must be amended in the light of this finding and, crucially, it is shown how in the vicinity of singularities in the wavenumber domain the procedure of matched asymptotics reveals the existence of exponentiallysmall terms missed by the conventional analysis. The ramifications of such terms are that only symmetric or antisymmetric solutions bifurcate from the point of expansion P = P
Recent Problems from Uniform Asymptotic Analysis of Integrals In Particular in Connection with Tricomi's $Psi$function
, 1998
"... The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of TricomiCarlitz polynomials in terms of Hermite polynomials. We mention ..."
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The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of TricomiCarlitz polynomials in terms of Hermite polynomials. We mention other recent expansions for orthogonal polynomials that do not satisfy a differential equation, and for which methods based on integral representations produce powerful uniform asymptotic expansions.
Uniform Asymptotic Expansions of Integrals: A Selection of Problems
, 1995
"... On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several "standard" problems and exam ..."
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On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several "standard" problems and examples, in which known special functions (error functions, Airy functions, Bessel functions, etc.) are needed to construct uniform approximations. Finally, we discuss the recent interest and new insights in the Stokes phenomenon. An extensive bibliography on uniform asymptotic methods for integrals is given, together with references to recent papers on the Stokes phenomenon for integrals and related topics.
Connection formulae
, 1998
"... for the degenerated asymptotic solutions of the fourth Painlevé equation ..."
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