Exponential asymptotics and Stokes lines in a partial differential equation

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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主要な著者: Chapman, S, Mortimer, D
フォーマット: Journal article
出版事項: 2005
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author Chapman, S
Mortimer, D
author_facet Chapman, S
Mortimer, D
author_sort Chapman, S
collection OXFORD
description 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.
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spelling oxford-uuid:fead6bdb-d66a-4c9b-b773-48568d79f78c2022-03-27T13:38:25ZExponential asymptotics and Stokes lines in a partial differential equationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fead6bdb-d66a-4c9b-b773-48568d79f78cMathematical Institute - ePrints2005Chapman, SMortimer, DA 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.
spellingShingle Chapman, S
Mortimer, D
Exponential asymptotics and Stokes lines in a partial differential equation
title Exponential asymptotics and Stokes lines in a partial differential equation
title_full Exponential asymptotics and Stokes lines in a partial differential equation
title_fullStr Exponential asymptotics and Stokes lines in a partial differential equation
title_full_unstemmed Exponential asymptotics and Stokes lines in a partial differential equation
title_short Exponential asymptotics and Stokes lines in a partial differential equation
title_sort exponential asymptotics and stokes lines in a partial differential equation
work_keys_str_mv AT chapmans exponentialasymptoticsandstokeslinesinapartialdifferentialequation
AT mortimerd exponentialasymptoticsandstokeslinesinapartialdifferentialequation