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A New Extended Jacobi Elliptic Function Expansion Method and Its Application to the Generalized Shallow Water Wave Equation

Author

Listed:
  • Yafeng Xiao
  • Haili Xue
  • Hongqing Zhang

Abstract

With the aid of symbolic computation, a new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi elliptic functions, are very effective to uniformly construct more new exact periodic solutions in terms of Jacobi elliptic function solutions of nonlinear partial differential equations. As an application of the method, we choose the generalized shallow water wave (GSWW) equation to illustrate the method. As a result, we can successfully obtain more new solutions. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.

Suggested Citation

  • Yafeng Xiao & Haili Xue & Hongqing Zhang, 2012. "A New Extended Jacobi Elliptic Function Expansion Method and Its Application to the Generalized Shallow Water Wave Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:896748
    DOI: 10.1155/2012/896748
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    References listed on IDEAS

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    2. Chen, Yong & Wang, Qi & Li, Biao, 2005. "Elliptic equation rational expansion method and new exact travelling solutions for Whitham–Broer–Kaup equations," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 231-246.
    3. He, Ji-Huan & Abdou, M.A., 2007. "New periodic solutions for nonlinear evolution equations using Exp-function method," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1421-1429.
    4. Petter A. Clarkson & Elizabeth L. Mansfield, 1995. "Symmetry Reductions and Exact Solutions of Shallow Water Wave Equations," Springer Books, in: Michiel Hazewinkel & Hans W. Capel & Eduard M. de Jager (ed.), KdV ’95, pages 245-276, Springer.
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