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Solving the one-loop soliton solution of the Vakhnenko equation by means of the Homotopy analysis method

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  • Wu, Yongyan
  • Wang, Chun
  • Liao, Shi-Jun

Abstract

A powerful, easy-to-use analytic technique for nonlinear problems, namely the Homotopy analysis method, is applied to solve the Vakhnenko equation, a nonlinear equation with loop soliton solutions governing the propagation of high-frequency waves in a relaxing medium. By means of the transformation of independent variables, an analysis one-loop soliton solution expressed by a series of exponential functions is obtained, which agrees well with the exact solution. This indicates the validity and great potential of the Homotopy analysis method in solving complicated solitary wave problems.

Suggested Citation

  • Wu, Yongyan & Wang, Chun & Liao, Shi-Jun, 2005. "Solving the one-loop soliton solution of the Vakhnenko equation by means of the Homotopy analysis method," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1733-1740.
    • Handle: RePEc:eee:chsofr:v:23:y:2005:i:5:p:1733-1740
    DOI: 10.1016/j.chaos.2004.06.081
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    Cited by:

    1. Sajid, M. & Hayat, T., 2008. "The application of homotopy analysis method to thin film flows of a third order fluid," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 506-515.
    2. Allan, Fathi M., 2009. "Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1744-1752.
    3. Liu, Yinping & Li, Zhibin, 2009. "The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 1-8.
    4. El-Nahhas, A., 2009. "Analytic approximations for the one-loop soliton solution of the Vakhnenko equation," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2257-2264.
    5. Alomari, A.K. & Noorani, M.S.M. & Nazar, R., 2009. "On the homotopy analysis method for the exact solutions of Helmholtz equation," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1873-1879.
    6. Jin-Shun, Feng & Ming-Pu, Guo & Deyou, Yuan, 2009. "The presentation of explicit analytical solutions of a class of nonlinear evolution equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2422-2428.
    7. Sajid, M. & Hayat, T., 2009. "The application of homotopy analysis method for MHD viscous flow due to a shrinking sheet," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1317-1323.

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