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

By S.J. Chapman, C.J. Howls, J.R. King and A.B. Olde Daalhuis

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

Topics: QA
Year: 2007
OAI identifier: oai:eprints.soton.ac.uk:48078
Provided by: e-Prints Soton
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