American Institute of Mathematical Sciences

August  2019, 39(8): 4455-4469. doi: 10.3934/dcds.2019182

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

 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  July 2018 Revised  October 2018 Published  May 2019

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.

Citation: Günther Hörmann, Hisashi Okamoto. Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4455-4469. doi: 10.3934/dcds.2019182
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References:
The solution from data1. $0 \le t \le 0.65$
data2
traveling wave $U$; $c = 0.025, 0.0255, 0.026, 0.0269$
The time dependent solution with the traveling wave as the initial data
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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