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Why is a shock not a caustic? The higher order Stokes phenomenon and smoothed shock formation

Why is a shock not a caustic? The higher order Stokes phenomenon and smoothed shock formation
Why is a shock not a caustic? The higher order Stokes phenomenon and smoothed shock formation
The formation of shocks in waves of advance in nonlinear partial differential equations is a well-explored problem and has been studied using many different techniques. In this paper we demonstrate how an exponential-asymptotic approach can be used to characterise completely the shock formation in a nonlinear PDE and so resolve an apparent paradox concerning the asymptotic modelling of shock formation. In so doing, we find that the recently discovered higher-order Stokes phenomenon plays a significant, previously unrealised, role in the asymptotic analysis of shocks. For the purposes of clarity, Burgers’ equation is used as a pedagogical example, but the techniques illustrated are more generally applicable.
asymptotics, exponential asymptotics, asymptotics beyond all orders, stokes phenomenon, higher order stokes phenomenon, nonlinear PDEs, shocks
0951-7715
2425-2452
Chapman, S.J.
2fb8f0b2-56ab-4ac7-8518-35abf1047cf9
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
King, J.R.
97bc791f-b608-4adb-a69a-ae992574b7b2
Olde Daalhuis, A.B.
d2254863-03c9-4e12-aee7-2855b60dc933
Chapman, S.J.
2fb8f0b2-56ab-4ac7-8518-35abf1047cf9
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
King, J.R.
97bc791f-b608-4adb-a69a-ae992574b7b2
Olde Daalhuis, A.B.
d2254863-03c9-4e12-aee7-2855b60dc933

Chapman, S.J., Howls, C.J., King, J.R. and Olde Daalhuis, A.B. (2007) Why is a shock not a caustic? The higher order Stokes phenomenon and smoothed shock formation. Nonlinearity, 20 (10), 2425-2452. (doi:10.1088/0951-7715/20/10/009).

Record type: Article

Abstract

The formation of shocks in waves of advance in nonlinear partial differential equations is a well-explored problem and has been studied using many different techniques. In this paper we demonstrate how an exponential-asymptotic approach can be used to characterise completely the shock formation in a nonlinear PDE and so resolve an apparent paradox concerning the asymptotic modelling of shock formation. In so doing, we find that the recently discovered higher-order Stokes phenomenon plays a significant, previously unrealised, role in the asymptotic analysis of shocks. For the purposes of clarity, Burgers’ equation is used as a pedagogical example, but the techniques illustrated are more generally applicable.

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Submitted date: 27 March 2007
Published date: 13 September 2007
Keywords: asymptotics, exponential asymptotics, asymptotics beyond all orders, stokes phenomenon, higher order stokes phenomenon, nonlinear PDEs, shocks

Identifiers

Local EPrints ID: 48078
URI: http://eprints.soton.ac.uk/id/eprint/48078
ISSN: 0951-7715
PURE UUID: 7a5c64f7-7f35-4d44-8c7e-99f4a2395629
ORCID for C.J. Howls: ORCID iD orcid.org/0000-0001-7989-7807

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Date deposited: 17 Sep 2007
Last modified: 09 Jan 2022 03:03

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Contributors

Author: S.J. Chapman
Author: C.J. Howls ORCID iD
Author: J.R. King
Author: A.B. Olde Daalhuis

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