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Uni-directional waves over slowly varying bottom. I: Derivation of a KdV- type of equation. (English) Zbl 0819.76012

(From the authors’ abstract.) The exact equations for surface waves over an uneven bottom can be formulated as a Hamiltonian system, with the total energy of the fluid as Hamiltonian. If the bottom variations are over a length scale \(L\) that is longer than the characteristic wavelength \(\ell\), the waves approximating the kinetic energy for the case of “rather long and rather low” gives equations of Boussinesq type. If in the case of an even bottom one restricts further to uni-directional waves, the Korteweg-de Vries (KdV) equation is obtained. For slowly varying bottom this uni-directionalization will be studied in detail in this part I, in a very direct way which is simpler than other derivations found in the literature. The surface elevation is shown to be described by an equation of forced KdV-type. The modification of the obtained KdV- equation shares the property of the standard KdV-equation and has a Hamiltonian structure, but now the structure map depends explicitly on the spatial variable through the bottom topography.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53 KdV equations (Korteweg-de Vries equations)
86A05 Hydrology, hydrography, oceanography
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