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

  • Jerry L. Bona (a1) and Ronald Smith (a1)

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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.

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(1)Benjamin, T. B. Lectures on nonlinear wave motion. Lectures in Applied Mathematics, vol. 15 (edited by Newell, A. C., American Mathematical Society, 1974).
(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.
(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.
(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.
(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.
(6)Hardy, G. H., Littlewood, J. E. and Pólya, G.Inequalities (2nd edition, Cambridge University Press, 1952).
(7)Long, R. R.The initial-value problem for long waves of finite amplitude. J. Fluid Mech. 20 (1964), 161.
(8)Madsen, O. S. and Mei, C. C.The transformation of a solitary wave over an uneven bottom. J. Fluid Mech. 39 (1969), 781.
(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).
(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.
(11)Yosida, K.Functional analysis (3rd edition, Springer-Verlag, 1971).

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