A dispersive model equation is considered, which has been proposed by Whitham [Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974] as a shallow water model, and which can also be seen as a caricature of two-species Euler--Poisson problems. A number of formal properties as well as similarities to other dispersive equations are derived. A travelling wave analysis and some numerical tests are carried out. The equation features wave breaking in finite time. A local existence result for smooth solutions and a global existence result for weak entropy solutions are proved. Finally, a small dispersion limit is carried out for situations where the solution of the limiting equation is smooth.
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Schmeiser et al. (2004) studied this question.
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