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 completely characterize the shock formation in a...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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2007
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| _version_ | 1797106431288147968 |
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| author | Chapman, S Howls, C King, JR Olde Daalhuis, AB |
| author_facet | Chapman, S Howls, C King, JR Olde Daalhuis, AB |
| author_sort | Chapman, S |
| collection | OXFORD |
| description | 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 completely characterize the shock formation in a nonlinear partial differential equation 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 unrealized, 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. © 2007 IOP Publishing Ltd and London Mathematical Society. |
| first_indexed | 2024-03-07T07:02:21Z |
| format | Journal article |
| id | oxford-uuid:ffec0be3-9d56-4f43-b814-035e86e5680e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:02:21Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:ffec0be3-9d56-4f43-b814-035e86e5680e2022-03-27T13:48:41ZWhy is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ffec0be3-9d56-4f43-b814-035e86e5680eEnglishSymplectic Elements at Oxford2007Chapman, SHowls, CKing, JROlde Daalhuis, ABThe 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 completely characterize the shock formation in a nonlinear partial differential equation 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 unrealized, 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. © 2007 IOP Publishing Ltd and London Mathematical Society. |
| spellingShingle | Chapman, S Howls, C King, JR Olde Daalhuis, AB Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation |
| title | Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation |
| title_full | Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation |
| title_fullStr | Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation |
| title_full_unstemmed | Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation |
| title_short | Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation |
| title_sort | why is a shock not a caustic the higher order stokes phenomenon and smoothed shock formation |
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