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A mathematical model for long waves generated by wavemakers in non-linear dispersive systems

Published online by Cambridge University Press:  24 October 2008

J. L. Bona
Affiliation:
Fluid Mechanics Research Institute, University of Essex
P. J. Bryant
Affiliation:
Fluid Mechanics Research Institute, University of Essex

Extract

An initial-boundary-value problem for the equation

is considered for x, t ≥ 0. This system is a model for long water waves of small but finite amplitude, generated in a uniform open channel by a wavemaker at one end. It is shown that, in contrast to an alternative, more familiar model using the Korteweg–deVries equation, the solution of (a) has good mathematical properties: in particular, the problem is well set in Hadamard's classical sense that solutions corresponding to given initial data exist, are unique, and depend continuously on the specified data.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

(1)Benjamin, T. B., Bona, J. L. & Mahony, J. J.Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London, Ser. A 272 (1972), 47.CrossRefGoogle Scholar
(2)Korteweg, D. J. & De Vries, G.On the change in form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Philos. Mag. (5) 39 (1895), 422.CrossRefGoogle Scholar
(3)Lions, J. L. & Magenes, E.Problèmes aux limites non homogènes et applications, vol. 1 (Paris, Dunod, 1968).Google Scholar
(4)Treves, F.Topological vector spaces, distributions and kernels (New York, Academic Press, 1967).Google Scholar
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