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Periodic cusp wave solutions and single-solitons for the b-equation

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  • Guo, Boling
  • Liu, Zhengrong

Abstract

In this paper, we employ the bifurcation method of dynamical systems and the numerical simulation approach of differential equations to study periodic cusp wave solutions and single-solitons for the b-equationut-uxxt+(b+1)uux=buxuxx+uuxxxwith b>1. The explicit representations of periodic cusp waves and the implicit expressions of single-solitons are obtained. Further, we show that the limits of both periodic cusp waves and single-solitons are peakons which possess explicit expression u=ce−∣x−ct∣. As corollary, the single-solitons equations of the Camassa–Holm equation and the Degasperis–Procesi equation are given. Our theoretical derivations are identical with the numerical simulations.

Suggested Citation

  • Guo, Boling & Liu, Zhengrong, 2005. "Periodic cusp wave solutions and single-solitons for the b-equation," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1451-1463.
    • Handle: RePEc:eee:chsofr:v:23:y:2005:i:4:p:1451-1463
    DOI: 10.1016/j.chaos.2004.06.062
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    Cited by:

    1. Yu, Liqin & Tian, Lixin & Wang, Xuedi, 2006. "The bifurcation and peakon for Degasperis–Procesi equation," Chaos, Solitons & Fractals, Elsevier, vol. 30(4), pages 956-966.
    2. 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.
    3. Xu, Chuanhai & Tian, Lixin, 2009. "The bifurcation and peakon for K(2,2) equation with osmosis dispersion," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 893-901.
    4. Chen, Can & Tang, Minying, 2006. "A new type of bounded waves for Degasperis–Procesi equation," Chaos, Solitons & Fractals, Elsevier, vol. 27(3), pages 698-704.

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