When is a Stokes line not a Stokes line?
When is a Stokes line not a Stokes line?
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.
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, why some Stokes lines are irrelevant to a given problem and why it is indispensible 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 linear partial differential equations.
Subsequently we apply these techniques to Burgers equation, a non-linear partial differential equation developed to model turbulent fluid flow. We find that the higher order Stokes phenomenon plays a major, yet very subtle role in the smoothed shock wave formation of this equation.
Langman, Philip J.
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December 2005
Langman, Philip J.
b008233f-0d6b-4300-bcc9-67a296de825c
Howls, C.J.
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Langman, Philip J.
(2005)
When is a Stokes line not a Stokes line?
University of Southampton, School of Mathematics, Doctoral Thesis, 135pp.
Record type:
Thesis
(Doctoral)
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.
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, why some Stokes lines are irrelevant to a given problem and why it is indispensible 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 linear partial differential equations.
Subsequently we apply these techniques to Burgers equation, a non-linear partial differential equation developed to model turbulent fluid flow. We find that the higher order Stokes phenomenon plays a major, yet very subtle role in the smoothed shock wave formation of this equation.
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Published date: December 2005
Organisations:
University of Southampton, Mathematical Sciences
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Local EPrints ID: 336279
URI: http://eprints.soton.ac.uk/id/eprint/336279
PURE UUID: 4e5928e7-d195-420b-a479-2027641b2449
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Date deposited: 20 Mar 2012 16:46
Last modified: 11 Dec 2021 03:34
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Author:
Philip J. Langman
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