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Wave breaking for a periodic shallow water equation. (English) Zbl 1042.35060
Summary: The focus of this paper is on the blow-up of a recently derived one-dimensional shallow water equation which is formally integrable and can be obtained by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime. Some new criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data are obtained for the periodic case.

MSC:
35Q35PDEs in connection with fluid mechanics
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction