Abstract
Bona and Smith (6) have suggested that the coupled system of equationshas the same formal justification as other Boussinesq-type models for the two-way propagation of one-dimensional water waves of small but finite amplitude in a channel with a flat bottom. The variables u and η represent the velocity and elevation of the free surface, respectively. Using the energy invariantthey show that for a restricted, but nevertheless physically relevant, class of initial data, the system (1·1) has solutions which exist for all time, and that in such circumstances the wave height is bounded solely in terms of the initial data.
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