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Model Equations and Traveling Wave Solutions for Shallow-Water Waves with the Coriolis Effect

  • Guilong Gui
  • Yue Liu
  • Ting Luo
Article
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Abstract

In the present study, we start by formally deriving the simplified phenomenological models of long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the Earth’s rotation. These new model equations are analogous to the Green–Naghdi equations, the first-order approximations of the KdV-, or BBM type, respectively. We then justify rigorously that in the long-wave limit, unidirectional solutions of a class of KdV- or BBM type are well approximated by the solutions of the Camassa–Holm equation in a rotating setting. The modeling and analysis of those mathematical models then illustrate that the Coriolis forcing in the propagation of shallow-water waves can not be neglected. Indeed, the CH-approximation with the Coriolis effect captures stronger nonlinear effects than the nonlinear dispersive rotational KdV type. Furthermore, we demonstrate nonexistence of the Camassa–Holm-type peaked solution and classify various localized traveling wave solutions to the Camassa–Holm equation with the Coriolis effect depending on the range of the rotation parameter.

Keywords

Coriolis effect Rotation-Camassa–Holm equation Rotation-KdV equation Rotation-Green–Naghdi equations Solitary waves 

Mathematics Subject Classification

35Q53 35B30 35G25 

Notes

Acknowledgements

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

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China
  2. 2.Chongqing UniversityChongqingPeople’s Republic of China
  3. 3.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA
  4. 4.Institute of Applied Physics and Computational MathematicsBeijingPeople’s Republic of China

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