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Bifurcations of travelling wave solutions in a new integrable equation with peakon and compactons

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  • Shen, Jianwei
  • Xu, Wei
  • Li, Wei

Abstract

Degasperis and Procesi applied the method of asymptotic integrability and obtain Degasperis–Procesi equation. They showed that it has peakon solutions, which has a discontinuous first derivative at the wave peak, but they did not explain the reason that the peakon solution arises. In this paper, we study these non-smooth solutions of the generalized Degasperis–Procesi equation ut−utxx+(b+1)uux=buxuxx+uuxxx, show the reason that the non-smooth travelling wave arise and investigate global dynamical behavior and obtain the parameter condition under which peakon, compacton and another travelling wave solutions engender. Under some parameter condition, this equation has infinitely many compacton solutions. Finally, we give some explicit expression of peakon and compacton solutions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:27:y:2006:i:2:p:413-425
    DOI: 10.1016/j.chaos.2005.04.020
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    Cited by:

    1. 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.
    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. 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.

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