IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v26y2005i5p1309-1316.html
   My bibliography  Save this article

Explicit solutions of the Camassa–Holm equation

Author

Listed:
  • Parkes, E.J.
  • Vakhnenko, V.O.

Abstract

Explicit travelling-wave solutions of the Camassa–Holm equation are sought. The solutions are characterized by two parameters. For propagation in the positive x-direction, both periodic and solitary smooth-hump, peakon, cuspon and inverted-cuspon waves are found. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. Some composite wave solutions of the Degasperis–Procesi equation are given in an appendix.

Suggested Citation

  • Parkes, E.J. & Vakhnenko, V.O., 2005. "Explicit solutions of the Camassa–Holm equation," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1309-1316.
    • Handle: RePEc:eee:chsofr:v:26:y:2005:i:5:p:1309-1316
    DOI: 10.1016/j.chaos.2005.03.011
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077905002651
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2005.03.011?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. Kalisch, Henrik & Lenells, Jonatan, 2005. "Numerical study of traveling-wave solutions for the Camassa–Holm equation," Chaos, Solitons & Fractals, Elsevier, vol. 25(2), pages 287-298.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. Qiao, Zhijun & Liu, Liping, 2009. "A new integrable equation with no smooth solitons," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 587-593.
    4. Parkes, E.J., 2008. "Some periodic and solitary travelling-wave solutions of the short-pulse equation," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 154-159.
    5. Parker, A., 2007. "Cusped solitons of the Camassa–Holm equation. I. Cuspon solitary wave and antipeakon limit," Chaos, Solitons & Fractals, Elsevier, vol. 34(3), pages 730-739.

    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. A. A. Alderremy & Hassan Khan & Rasool Shah & Shaban Aly & Dumitru Baleanu, 2020. "The Analytical Analysis of Time-Fractional Fornberg–Whitham Equations," Mathematics, MDPI, vol. 8(6), pages 1-14, June.
    2. Xianguo Geng & Ruomeng Li, 2019. "On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation," Mathematics, MDPI, vol. 7(10), pages 1-23, October.
    3. Shijie Zeng & Yaqing Liu, 2023. "The Whitham Modulation Solution of the Complex Modified KdV Equation," Mathematics, MDPI, vol. 11(13), pages 1-18, June.
    4. Yuheng Jiang & Yu Tian & Yao Qi, 2024. "Solitary Wave Solutions of a Hyperelastic Dispersive Equation," Mathematics, MDPI, vol. 12(4), pages 1-10, February.
    5. Katrin Grunert & Audun Reigstad, 2021. "Traveling waves for the nonlinear variational wave equation," Partial Differential Equations and Applications, Springer, vol. 2(5), pages 1-21, October.
    6. Parkes, E.J., 2008. "Some periodic and solitary travelling-wave solutions of the short-pulse equation," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 154-159.
    7. Hendrik Ranocha & Manuel Quezada Luna & David I. Ketcheson, 2021. "On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations," Partial Differential Equations and Applications, Springer, vol. 2(6), pages 1-26, December.
    8. 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.
    9. Yin, Jiuli & Tian, Lixin, 2009. "Stumpons and fractal-like wave solutions to the Dullin–Gottwald–Holm equation," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 643-648.

    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:26:y:2005:i:5:p:1309-1316. 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.