IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v76y2015icp40-46.html
   My bibliography  Save this article

Orbital stability and dynamical behaviors of solitary waves for the Camassa–Holm equation with quartic nonlinearity

Author

Listed:
  • Yin, Jiuli
  • Xing, Qianqian
  • Tian, Lixin

Abstract

In this paper we prove that the Camassa–Holm equation with quartic nonlinearity is non-integrable via the Painlevé method. The orbital stability of solitary waves for this equation is investigated by constructing a functional extremum problem. This result demonstrates that the resulting solitary wave is unstable when its speed lies in the narrow region of the critical value that connects with the bifurcation condition. In contrast when the speed surpasses the narrow region, the solitary wave is stable. In addition, the stable solitary wave turns into a chaotic state when is driven externally. If a damping term controller is added to the perturbed equation, the solitary wave can also propagate stably under a certain condition. Finally our numerical results show that the perturbed equation is not well controlled when a certain resonant-frequency occurs and is well controlled with a smaller wave speed as well as a higher nonlinear convection.

Suggested Citation

  • Yin, Jiuli & Xing, Qianqian & Tian, Lixin, 2015. "Orbital stability and dynamical behaviors of solitary waves for the Camassa–Holm equation with quartic nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 40-46.
    • Handle: RePEc:eee:chsofr:v:76:y:2015:i:c:p:40-46
    DOI: 10.1016/j.chaos.2015.03.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077915000831
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2015.03.002?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Shen, Jianwei & Xu, Wei & Li, Wei, 2006. "Bifurcations of travelling wave solutions in a new integrable equation with peakon and compactons," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 413-425.
    2. Yan, Zhenya, 2003. "Painlevé analysis, auto-Bäcklund transformations and exact solutions for a simplified model for reacting mixtures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 326(3), pages 344-359.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Abbasbandy, S., 2009. "Solitary wave solutions to the modified form of Camassa–Holm equation by means of the homotopy analysis method," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 428-435.
    2. Abbasbandy, S. & Parkes, E.J., 2008. "Solitary smooth hump solutions of the Camassa–Holm equation by means of the homotopy analysis method," Chaos, Solitons & Fractals, Elsevier, vol. 36(3), pages 581-591.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:76:y:2015:i:c:p:40-46. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.