Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation

Günther Hörmann; Hisashi Okamoto

doi:10.3934/dcds.2019182

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2019, Volume 39, Issue 8: 4455-4469. Doi: 10.3934/dcds.2019182

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Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation

Günther Hörmann^1, and
Hisashi Okamoto^2,

1.
Fakultät für Mathematik, Universität Wien, Austria
2.
Department of Mathematics, Gakushuin University, Tokyo, Japan

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received: June 30, 2018

Revised: September 30, 2018

Published: May 2019

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Abstract

Spatially periodic solutions of the Fornberg-Whitham equation are studied to illustrate the mechanism of wave breaking and the formation of shocks for a large class of initial data. We show that these solutions can be considered to be weak solutions satisfying the entropy condition. By numerical experiments, we show that the breaking waves become shock-wave type in the time evolution.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35D30.

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Figure 1. The solution from data1. $ 0 \le t \le 0.65 $

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Figure 2. data2

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Figure 3. traveling wave $ U $; $ c = 0.025, 0.0255, 0.026, 0.0269 $

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Figure 4. The time dependent solution with the traveling wave as the initial data

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Figure 5. The time dependent solution with $ d $ as in the second case of (12). The points $ ( u_{300}^n,u_{600}^n) $ with $ n $ corresponding to $ 0 \le t \le 300 $ are plotted

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