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IMA Journal of Applied Mathematics Advance Access published online on February 9, 2007
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxm002
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Explicit modulation equations, Reynolds averaging and the closure problem for the Korteweg-deVries–Burgers equation

Warren R. Smith{dagger}

School of Mathematics, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

{dagger} Email: smithwar{at}maths.bham.ac.uk

Received on 5 December 2005; Revision received 5 January 2007.
  Abstract

Original explicit modulation equations are determined for cnoidal waves of the Korteweg-deVries (KdV)–Burgers equation. This formal asymptotic analysis is used to demonstrate that there is no single partial differential equation for the leading-order mean velocity. The technique of Reynolds averaging is also employed to determine an equation for the mean velocity with the familiar closure problem being encountered. The Reynolds-averaged KdV–Burgers equation is shown to be a counterexample to the existence of a closure associated with a convective nonlinearity.

Keywords: strongly nonlinear analysis; Reynolds averaging.


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