Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves

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© 2003 The Japan Society of Fluid Mechanics and IOP Publishing Ltd
, , Citation Holger R Dullin et al 2003 Fluid Dyn. Res. 33 73 DOI 10.1016/S0169-5983(03)00046-7

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Author affiliations

1 Department of Mathematical Sciences, Loughborough University, Leicestershire, Loughborough, LE11 3TU, UK

2 Mathematics and Statistics, University of Sydney, NSW 2006, Australia

3 Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, MS B284, Los Alamos, NM 87545, USA

4 Mathematics Department, Imperial College of Science, Technology and Medicine, London SW7 2AZ, UK

5 Corresponding author

Dates

  1. Received 10 August 2002
  2. Accepted 23 January 2003

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1873-7005/33/1-2/73

Abstract

We derive the Camassa–Holm equation (CH) as a shallow water wave equation with surface tension in an asymptotic expansion that extends one order beyond the Korteweg–de Vries equation (KdV). We show that CH is asymptotically equivalent to KdV5 (the fifth-order integrable equation in the KdV hierarchy) by using the non-linear/non-local transformations introduced in Kodama (Phys. Lett. A 107 (1985a) 245; Phys. Lett. A 112 (1985b) 193; Phys. Lett. A 123 (1987) 276). We also classify its travelling wave solutions as a function of Bond number by using phase plane analysis. Finally, we discuss the experimental observability of the CH solutions.

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Communicated by T Mullin

10.1016/S0169-5983(03)00046-7