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A model for the two-way propagation of water waves in a channel

Published online by Cambridge University Press:  24 October 2008

Jerry L. Bona  and
Ronald Smith
Jerry L. Bona
Affiliation:
University of Chicago and University of Cambridge
Ronald Smith
Affiliation:
University of Chicago and University of Cambridge
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Extract

Global existence, uniqueness and regularity of solutions and continuous dependence of solutions on varied initial data are established for the initial-value problem for the coupled system of equations

This system has the same formal justification as a model for the two-way propagation of (one-dimensional) long waves of small but finite amplitude in an open channel of water of constant depth as other versions of the Boussinesq equations. A feature of the analysis is that bounds on the wave amplitude η are obtained which are valid for all time.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 1 , January 1976 , pp. 167 - 182
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Benjamin, T. B. Lectures on nonlinear wave motion. Lectures in Applied Mathematics, vol. 15 (edited by Newell, A. C., American Mathematical Society, 1974).Google Scholar
(2)Benjamin, T. B., Bona, J. L. and Mahony, J. J.Model equations for long waves in non-linear dispersive systems. Phil. Trans. Roy. Soc. (London), Ser. A 272 (1972), 47.Google Scholar
(3)Bona, J. L. and Smith, R.The initial-value problem for the Korteweg–de Vries equation. Phil. Trans. Roy. Soc. (London), Ser. A 278 (1975), 555.Google Scholar
(4)Boussinesq, M. J.Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal. Comptes Rendus de l'Academie des Sciences 73 (1871), 256.Google Scholar
(5)Byatt-Smith, J. G. B.An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J. Fluid Mech. 49 (1971), 625.CrossRefGoogle Scholar
(6)Hardy, G. H., Littlewood, J. E. and Pólya, G.Inequalities (2nd edition, Cambridge University Press, 1952).Google Scholar
(7)Long, R. R.The initial-value problem for long waves of finite amplitude. J. Fluid Mech. 20 (1964), 161.CrossRefGoogle Scholar
(8)Madsen, O. S. and Mei, C. C.The transformation of a solitary wave over an uneven bottom. J. Fluid Mech. 39 (1969), 781.CrossRefGoogle Scholar
(9)Peregrine, D. H. Equations for water waves and the approximations behind them. Waves on Beaches and Resulting Sediment Transport (edited by Meyer, R. E., Academic Press, 1972).Google Scholar
(10)Peregrine, D. H.Discussion of the paper ‘A numerical simulation of the undular hydraulic jump’ by M. B. Abbott and G. S. Rodenhuis. Journal of Hydraulic Research 12 (1974), 141.CrossRefGoogle Scholar
(11)Yosida, K.Functional analysis (3rd edition, Springer-Verlag, 1971).CrossRefGoogle Scholar