On the Modelling of Shallow-Water Waves with the Coriolis Effect

Abstract

Consideration herein is a rotation-Camassa–Holm-type equation, which can be derived as an asymptotic model for the propagation of long-crested shallow-water waves in the equatorial ocean regions with the weak Coriolis effect due to the Earth’s rotation, and is also related to the compressible hyperelastic rod model in the material science. This model equation has a formal Hamiltonian structure, and its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the solutions blow up in finite time in the sense of wave breaking. A refined analysis based on the local structure of the dynamics is performed to provide the wave-breaking phenomena. The effects of the Coriolis force caused by the Earth’s rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Finally, a sufficient condition for global strong solutions to the equation in some special case is given.