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Integral Bifurcation Method together with a Translation‐Dilation Transformation for Solving an Integrable 2‐Component Camassa‐Holm Shallow Water System

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  • Weiguo Rui
  • Yao Long

Abstract

An integrable 2‐component Camassa‐Holm (2‐CH) shallow water system is studied by using integral bifurcation method together with a translation‐dilation transformation. Many traveling wave solutions of nonsingular type and singular type, such as solitary wave solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution are obtained. Further more, their dynamic behaviors are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameter, that is to say, the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.

Suggested Citation

  • Weiguo Rui & Yao Long, 2012. "Integral Bifurcation Method together with a Translation‐Dilation Transformation for Solving an Integrable 2‐Component Camassa‐Holm Shallow Water System," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:736765
    DOI: 10.1155/2012/736765
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    References listed on IDEAS

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    1. Xianbin Wu & Weiguo Rui & Xiaochun Hong, 2012. "Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for K(m, n) Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    2. Xianbin Wu & Weiguo Rui & Xiaochun Hong, 2012. "Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for K ( m , n ) Equation," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-23, March.
    3. Weiguo Zhang & Xiang Li, 2011. "Approximate Damped Oscillatory Solutions for Generalized KdV-Burgers Equation and Their Error Estimates," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-26, September.
    4. Weiguo Zhang & Xiang Li, 2011. "Approximate Damped Oscillatory Solutions for Generalized KdV‐Burgers Equation and Their Error Estimates," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
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    Cited by:

    1. Yongsheng Mi & Chunlai Mu, 2013. "Well‐Posedness, Blow‐Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two‐Component Camassa‐Holm Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    2. Jianmei Zhang & Lixin Tian, 2013. "Wave‐Breaking Criterion for the Generalized Weakly Dissipative Periodic Two‐Component Hunter‐Saxton System," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    3. Weiguo Rui, 2013. "Exact Traveling Wave Solutions for a Nonlinear Evolution Equation of Generalized Tzitzéica‐Dodd‐Bullough‐Mikhailov Type," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).

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