A Nonlocal Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect

  • Guilong Gui
  • Yue LiuEmail author
  • Junwei Sun


In the present study a mathematical model of the equatorial water waves propagating mainly in one direction with the effect of Earth’s rotation is derived by the formal asymptotic procedures in the equatorial zone. Such a model equation is analogous to the Camassa–Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.


Coriolis effect Rotation-Camassa–Holm equation shallow water wave breaking 

Mathematics Subject Classification

35Q53 35B30 35G25 



The authors would like to thank the referees for constructive suggestions and comments. The work of Gui is supported in part by the NSF-China under the Grant Nos. 11571279, 11331005, and the Foundation FANEDD-201315. The work of Liu is supported in part by the Simons Foundation Grant-499875.

Compliance with ethical standards

Conflict of interest

The authors certify that they have no conflicts of interest concerning this work.


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Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA

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