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

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

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  • 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.

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

    Citation:

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

    Figure 2.  data2

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

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

    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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