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Asymptotics of Long Nonlinear Propagating Waves in a One-Dimensional Basin with Gentle Shores

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Abstract

The Cauchy problem for a one-dimensional (nonlinear) shallow water equations over a variable bottom \(D(x)\) is considered in an extended basin bounded from two sides by shores (where the bottom degenerates, \(D(a)=0\)), or by a shore and a wall. The short-wave asymptotics of the linearized system in the form of a propagating localized wave is constructed. After applying to the constructed functions a simple parametric or explicit change of variables proposed in recent papers (Dobrokhotov, Minenkov, Nazaikinsky, 2022 and Dobrokhotov, Kalinichenko, Minenkov, Nazaikinsky, 2023), we obtain the asymptotics of the original nonlinear problem. On the constructed families of functions, the ratio of the amplitude and the wavelength is studied for which hte wave does not collapse when running up to the shore.

DOI 10.1134/S1061920823040143

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Notes

  1. Tsunami waves and seiches, which are long as as compared to the depth of the reservoir, are at the same time short compared to the size of the basin.

  2. We need these expressions below as an ansatz for constructing asymptotics of the linear system (1.6) in the case of general bottom.

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Acknowledgments

The authors are grateful to Sergey Yu. Dobrokhotov for numerous discussions and support and to Vladimir E. Nazaikinskii for valuable advises. One of the author, D. S. Minenkov, also wants to express gratitude to his school teacher of physics Lidia P. Kocheshkova for cultivating physical intuition and the way of thinking.

Funding

The work was carried out within the framework of the Russian Science Foundation grant 21-11-00341.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia

    D.S. Minenkov & M.M. Votiakova

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  1. D.S. Minenkov
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  2. M.M. Votiakova
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Minenkov, D., Votiakova, M. Asymptotics of Long Nonlinear Propagating Waves in a One-Dimensional Basin with Gentle Shores. Russ. J. Math. Phys. 30, 621–642 (2023). https://doi.org/10.1134/S1061920823040143

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