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Exact and Explicit Solutions of Whitham-Broer-Kaup Equations in Shallow Water

Received: 2 September 2016     Accepted: 18 September 2016     Published: 17 October 2016
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Abstract

In this paper, a simple direct method is presented to find equivalence transformation of a nonlinear Whitham- Broer-Kaup equations. Applying this equivalence transformation, we can obtain the symmetry group theorem of the Whitham-Broer-Kaup equations and then derive series of new exact and explicit solutions of the Whitham-Broer-Kaup equations according to solutions of the previous references.

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 6)
DOI 10.11648/j.pamj.20160506.11
Page(s) 174-180
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Whitham-Broer-Kaup Equations, Direct Method, Equivalence Transformation, Symmetry, Explicit Solutions

References
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[9] X. Y. Jiao, H. Q. Zhang, An extended method and its application to Whitham-Broer-Kaup equation and two- dimensional perturbed KdV equation, Applied Mathematics and Computation, 172(2006)664-677.
[10] D. J. Kaup, A higher-order water-wave equation and the method for solving it, Progress of Theoretical Physics 54 (1975) 396–408.
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[13] P. J. Olver, Applications of Lie groups to differential equations, Springer-Verlag, 1993.
[14] J. W. Shen, W. Xu, Y. F. Jin, Bifurcation method and traveling wave solution to Whitham-Broer-Kaup equation, Applied Mathematics and Computation,171(2005)677-702.
[15] Sirendaoreji, A new auxiliary equation and exact travelling wave solutions of nonlinear equations, Physics Letters A, 356 (2006)124-130.
[16] G. B. Whitham, Variational methods and applications to water waves, Proceedings of the Royal Society of London, Series A 299 (1967) 6–25.
[17] F. D. Xie, Z. Y. Yan, H. Q Zhang, Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations, Physics Letters A, 285(2001)76-80.
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Cite This Article
  • APA Style

    Baodan Tian, Yanhong Qiu. (2016). Exact and Explicit Solutions of Whitham-Broer-Kaup Equations in Shallow Water. Pure and Applied Mathematics Journal, 5(6), 174-180. https://doi.org/10.11648/j.pamj.20160506.11

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    ACS Style

    Baodan Tian; Yanhong Qiu. Exact and Explicit Solutions of Whitham-Broer-Kaup Equations in Shallow Water. Pure Appl. Math. J. 2016, 5(6), 174-180. doi: 10.11648/j.pamj.20160506.11

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    AMA Style

    Baodan Tian, Yanhong Qiu. Exact and Explicit Solutions of Whitham-Broer-Kaup Equations in Shallow Water. Pure Appl Math J. 2016;5(6):174-180. doi: 10.11648/j.pamj.20160506.11

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  • @article{10.11648/j.pamj.20160506.11,
      author = {Baodan Tian and Yanhong Qiu},
      title = {Exact and Explicit Solutions of Whitham-Broer-Kaup Equations in Shallow Water},
      journal = {Pure and Applied Mathematics Journal},
      volume = {5},
      number = {6},
      pages = {174-180},
      doi = {10.11648/j.pamj.20160506.11},
      url = {https://doi.org/10.11648/j.pamj.20160506.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160506.11},
      abstract = {In this paper, a simple direct method is presented to find equivalence transformation of a nonlinear Whitham- Broer-Kaup equations. Applying this equivalence transformation, we can obtain the symmetry group theorem of the Whitham-Broer-Kaup equations and then derive series of new exact and explicit solutions of the Whitham-Broer-Kaup equations according to solutions of the previous references.},
     year = {2016}
    }
    

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    AB  - In this paper, a simple direct method is presented to find equivalence transformation of a nonlinear Whitham- Broer-Kaup equations. Applying this equivalence transformation, we can obtain the symmetry group theorem of the Whitham-Broer-Kaup equations and then derive series of new exact and explicit solutions of the Whitham-Broer-Kaup equations according to solutions of the previous references.
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Author Information
  • School of science, Southwest University of Science and Technology, Mianyang, China

  • School of science, Southwest University of Science and Technology, Mianyang, China

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