The Mathematical Institute, University of Oxford, Eprints Archive

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

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

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Official URL: http://stacks.iop.org/0951-7715/20/2425

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 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.

Item Type:Article
Subjects:O - Z > Partial differential equations
A - C > Approximations and expansions
Research Groups:Oxford Centre for Industrial and Applied Mathematics
ID Code:655
Deposited By:Jon Chapman
Deposited On:17 Sep 2007
Last Modified:20 Jul 2009 14:23

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