Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T21:15:33.803Z Has data issue: false hasContentIssue false
  • English
  • Français

On an asymptotic solution of the Korteweg–de Vries equation with slowly varying coefficients

Published online by Cambridge University Press:  29 March 2006

R. S. Johnson
R. S. Johnson
Affiliation:
School of Mathematics, University of Newcastle upon Tyne
Get access Rights & Permissions [Opens in a new window]

Abstract

The variable-coefficient Korteweg–de Vries equation \[ H_X + {\textstyle\frac{3}{2}}d^{-\frac{7}{4}}HH_{\xi} + {\textstyle\frac{1}{6}}\kappa d^{\frac{1}{2}}H_{\xi\xi\xi} = 0 \] with d = d(εX) is discussed for solitary-wave initial profiles. A straightforward asymptotic solution for ε → 0 is constructed and is shown to be non-uniform both ahead of and behind the solitary wave. The behaviour ahead is rectified by matching to the appropriate exponential form and, together with the use of conservation laws for the equation, the nature of the solution behind the solitary wave is discussed. This leads to the formulation of the solution in the oscillatory ‘tail’, which is again matched directly.

The results are applied to the development of the solitary wave into variable-depth water, and the predictions are compared with those obtained, for example, by Grimshaw (1970, 1971). Finally, the asymptotic behaviour of both the solitary wave and the oscillatory tail are assessed in the light of some numerical integrations of the equation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 60 , Issue 4 , 9 October 1973 , pp. 813 - 824
Copyright
© 1973 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ablowitz, M. J. 1971 Studies in Appl. Math. 50, 329–344.
Berezin, Y. A. & Karpman, V. I. 1964 Soviet Phys., J.E.T.P. 19, 1880–1890.
Grimshaw, R. 1970 J. Fluid Mech. 42, 639–656.
Grimshaw, R. 1971 J. Fluid Mech. 46, 611–622.
Johnson, R. S. 1972 J. Fluid Mech. 54, 81–91.
Johnson, R. S. 1973 Proc. Camb. Phil. Soc. 73, 183–203.
Kakutani, T., Ono, H., Taniuti, T. & Wei, C.-C. 1968 J. Phys. Soc. Japan, 24, 1159–1166.
Korteweg, D. J. & de Vries, G. 1895 Phil. Mag. 39, 422–443.
Madsen, O. S. & Mei, C. C. 1969 J. Fluid Mech. 39, 781–791.
Miura, R. M., Gardner, C. S. & Kruskal, M. D. 1968 J. Math. Phys. 9, 1204–1209.
Ostrovskiy, L. A. & Pelinovskiy, E. N. 1970 Atmos. Ocean. Phys. 6 (9), 934–939.
Tappert, F. & Zabusky, N. J. 1971 Phys. Rev. Lett. 27, 1774–1776.