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Three-dimensional nonlinear dispersive waves on shear flows. (English) Zbl 1145.76336

Summary: The Green-Naghdi equations describing three-dimensional water waves are considered. Assuming that transverse variations of the flow occur at a much shorter lengthscale than variations along the wave propagation direction, we derive simplified asymptotic equations from the Green-Naghdi model. For steady flows, we show that the approximate model reduces to a one-dimensional Hamiltonian system along each stream line. Exact solutions describing a wide class of free-boundary flows depending on several arbitrary functions of one argument are found. The numerical results showing different patterns of steady three-dimensional waves are presented.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53 KdV equations (Korteweg-de Vries equations)
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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References:

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