On the higher order Stokes phenomenon
On the higher order Stokes phenomenon
During the course of a Stokes phenomenon, an asymptotic expansion can change its form as a further series, prefactored by an exponentially small term and a Stokes multiplier, appears in the representation. The initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters. In this paper we introduce the concept of a 'higher-order Stokes phenomenon', at which a Stokes multiplier itself can change value. We show that the higher-order Stokes phenomenon can be used to explain the apparent sudden birth of Stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions. We provide an example of how the higher-order Stokes phenomenon can have important effects on the large-time behaviour of partial differential equations.
asymptotic expansions, hyperasymptotics, partial differential equations, steepest descent, Stokes phenomenon, turning points
2285-2303
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Langman, P.J.
b3d47d27-214b-478d-ba78-9c323f2f17ea
Daalhuis, A.B.O.
6b385a57-b024-4e2e-9f15-ceb2bfc5483a
2004
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Langman, P.J.
b3d47d27-214b-478d-ba78-9c323f2f17ea
Daalhuis, A.B.O.
6b385a57-b024-4e2e-9f15-ceb2bfc5483a
Howls, C.J., Langman, P.J. and Daalhuis, A.B.O.
(2004)
On the higher order Stokes phenomenon.
Proceedings of the Royal Society A, 460 (2048), .
(doi:10.1098/rspa.2004.1299).
Abstract
During the course of a Stokes phenomenon, an asymptotic expansion can change its form as a further series, prefactored by an exponentially small term and a Stokes multiplier, appears in the representation. The initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters. In this paper we introduce the concept of a 'higher-order Stokes phenomenon', at which a Stokes multiplier itself can change value. We show that the higher-order Stokes phenomenon can be used to explain the apparent sudden birth of Stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions. We provide an example of how the higher-order Stokes phenomenon can have important effects on the large-time behaviour of partial differential equations.
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Published date: 2004
Keywords:
asymptotic expansions, hyperasymptotics, partial differential equations, steepest descent, Stokes phenomenon, turning points
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Local EPrints ID: 29222
URI: http://eprints.soton.ac.uk/id/eprint/29222
ISSN: 1364-5021
PURE UUID: 04aa7537-11da-4383-bac2-ef79e6ee9fb7
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Date deposited: 11 May 2006
Last modified: 16 Mar 2024 03:13
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Author:
P.J. Langman
Author:
A.B.O. Daalhuis
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