The Cauchy problem for shallow water waves of large amplitude in Besov space

We considered the propagation of nonlinear shallow water waves in a narrow channel presenting a fork. We aimed at computing the coupling conditions for a 1D effective model, using 2D simulations and an analysis based on the conservation laws. For small amplitudes, this analysis justifies the well-known Stoker interface conditions, so that the coupling does not depend on the angle of the fork. We also find this in the numerical solution. Large amplitude solutions in a symmetric fork also tend to follow Stoker’s relations, due to the symmetry constraint. For non symmetric forks, 2D effects dominate so that it is necessary to understand the flow inside the fork. However, even then, conservation laws give some insight in the dynamics.

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SHALLOW WATER WAVES A COMPARISON OF THEORIES AND EXPERIMENTS

Coastal Engineering Proceedings ◽

10.9753/icce.v11.7 ◽

1968 ◽

Vol 1 (11) ◽

pp. 7 ◽

Cited By ~ 8

Author(s):

Bernard Le Mahaute ◽

David Divoky ◽

Albert Lin

Keyword(s):

Velocity Field ◽

Shallow Water ◽

Water Waves ◽

Large Amplitude ◽

Wave Theory ◽

Engineering Practice ◽

Cnoidal Wave ◽

Shallow Water Waves ◽

Series Of Experiments ◽

Water Depths

A series of experiments were performed to determine the velocity field and other characteristics of large amplitude shallow water waves. The experimental results were compared with the predictions of a variety of wave theories including those commonly used in engineering practice. While no theory was found exceptionally accurate, the cnoidal wave theory of Keulegan and Patterson appears most adequate for the range of wavelengths and water depths studied.

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Camassa-Holm equation on shallow water wave equation

International Journal of Scientific Research and Management ◽

10.18535/ijsrm/v5i12.23 ◽

2017 ◽

Vol 5 (12) ◽

pp. 7758-7764

Author(s):

Sh. Hajrulla, L. Bezati, F. Hoxha

Keyword(s):

Cauchy Problem ◽

Wave Equation ◽

Weak Solution ◽

Shallow Water ◽

Water Waves ◽

Water Wave ◽

Kdv Equation ◽

Shallow Water Wave ◽

Holm Equation ◽

The Cauchy Problem

In this paper we can consider the problem of week solutions for the general shallow water wave equation. In the first part of this paper, we deal to the well-known Kdv equation. We obtain the Camassa-Holm equation in particular. Both of them describe unidirectional shallow water waves equation. Moreover, all these equations have a bi-Hamiltonian structure, they are completely integrable, they have infinitely many conserved quantities. From a mathematical point of view the Camassa-Holm equation is well studied. In the second part of this paper, we obtain a global weak solution as a limit of approximation under the assumption Some concepts related to high dimensional spaces are considered. Then the Cauchy problem is considered. It has an admissible weak solution to the Cauchy problem for Existence, uniqueness, and basic energy estimate on this approximate solution sequence are given in some lemmas. Finally, the main theorem and the proof is given

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