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.
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Acknowledgements
The authors would like to express their gratitude to the referees for their useful and constructive comments. This work was initiated while Huang was visiting Department of Mathematics, University of Texas at Arlington as a Ph.D visiting student during the year 2018–2019, who would like to thank the department for its warm hospitality and support. The work of Chen is supported in part by the Global Change Research Program of China (No. 2015CB953904) and the National Natural Science Foundation of China under grants 11675054 and 11435005. The work of Huang is partially supported by the East China Normal University postgraduate study abroad grant. The work of Liu is partially supported by the Simons Foundation Under Grant 499875.
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Communicated by Darryl D. Holm.
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Chen, Y., Huang, L. & Liu, Y. On the Modelling of Shallow-Water Waves with the Coriolis Effect. J Nonlinear Sci 30, 93–135 (2020). https://doi.org/10.1007/s00332-019-09569-w
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DOI: https://doi.org/10.1007/s00332-019-09569-w