Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T22:42:03.939Z Has data issue: false hasContentIssue false
  • English
  • Français

Shelves and the Korteweg-de Vries equation

Published online by Cambridge University Press:  19 April 2006

C. J. Knickerbocker  and
Alan C. Newell
C. J. Knickerbocker
Affiliation:
Department of Mathematics and Computer Science, Clarkson College, Potsdam, NY 13676
Alan C. Newell
Affiliation:
Department of Mathematics and Computer Science, Clarkson College, Potsdam, NY 13676
Get access Rights & Permissions [Opens in a new window]

Abstract

An extension of the analytical results of Kaup & Newell (1978) concerning the effect of a perturbation on a solitary wave of the Korteweg–de Vries equation is given and numerical studies are conducted to verify the conclusions. In all cases, the numerical results agree with the results predicted by the theory. The most striking feature of the perturbed flow is the presence of a shelf in the lee of the solitary wave whose role is to absorb (provide) the extra mass which is created (depleted) by the perturbation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 98 , Issue 4 , 26 June 1980 , pp. 803 - 818
Copyright
© 1980 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

Djordjevic, V. D. & Redekopp, L. G. 1978 The fission and disintegration of internal solitary waves moving over two-dimensional topography. J. Phys. Oceanog. 8, 1016–1024.Google Scholar
Flaschka, H. & Newell, A. C. 1975 Dynamical Systems, Theory and Applications (ed. J. Moser), Lecture Notes in Physics, vol. 38, p. 335. Springer.
Gardner, C. S. 1971 The Korteweg-de Vries equation as a Hamiltonian system. J. Math. Phys. 12, 1548–1551.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving the Korteweg de Vries equation. Phys. Rev. Lett. 19, 1095–1097.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1974 Korteweg-de Vries equation and generalizations. VI. Methods for exact solution. Comm. Pure Appl. Math. 27, 97–133.Google Scholar
Grimshaw, R. 1970 The solitary wave in water of variable depth. J. Fluid Mech. 42, 639–656.Google Scholar
Grimshaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611–622.Google Scholar
Johnson, R. S. 1973a Asymptotic solution of the Korteweg-de Vries with slowly varying coefficients. J. Fluid Mech. 60, 813–824.Google Scholar
Johnson, R. S. 1973b On the development of a solitary wave moving over an uneven bottom. Proc. Camb. Phil. Soc. 73, 183–203.Google Scholar
Kakutani, T. 1971 Effects of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30, 272–276.Google Scholar
Karpman, V. I. & Maslov, E. M. 1977 A perturbation theory for the Korteweg-de Vries equation. Phys. Lett. A 60, 307–308.Google Scholar
Kaup, D. J. & Newell, A. C. 1978 Solitons as particles and oscillators and in slowly varying media: a singular perturbation theory. Proc. Roy. Soc. A 361, 413–446.Google Scholar
Ko, K. & Kuehl, H. H. 1978 Korteweg-de Vries soliton in a slowly varying medium. Phys. Rev. Lett. 4, 233–236.Google Scholar
Leibovich, S. & Randall, J. D. 1971 Dissipative effects on nonlinear waves in rotating fluids. Phys. Fluids 14, 2559–2561.Google Scholar
Leibovich, S. & Randall, J. D. 1973 Amplification and decay of long nonlinear waves. J. Fluid Mech. 58, 481–493.Google Scholar
Miles, J. W. 1979 On the Korteweg-de Vries equation for a gradually varying channel. J. Fluid Mech. 91, 181–190.Google Scholar
Newell, A. C. 1978 Soliton perturbation and nonlinear focussing, symposium on nonlinear structure and dynamics in condensed matter. In Solid State Physics, vol. 8, pp. 52–68. Oxford University Press.
Newell, A. C. 1980 The inverse scattering transform. Solitons (ed. R. K. Bullough) Topics in Current Physics, vol. 17. Springer.
Ott, E. & Sudan, R. N. 1970 Damping of solitary waves. Phys. Fluids 13, 1432–1434.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Tappert, F. & Zabusky, N. J. 1971 Gradient-induced fission of solitons. Phys. Rev. Lett. 27, 1774–1776.Google Scholar
Vliegenthart, A. C. 1971 On finite difference methods for the Korteweg-de Vries equation. J. Engng Math. 5, 137–155.Google Scholar
Zabusky, N. J. & Kruskal, M. D. 1965 Interactions of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243.Google Scholar
Zakharov, V. E. & Faddeev, L. D. 1971 The Korteweg-de Vries equation. A completely integrable Hamiltonian system. Anal. Priloz. 5, 18.Google Scholar