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New Types of Nonlinear Waves and Bifurcation Phenomena in Schamel‐Korteweg‐de Vries Equation

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  • Yun Wu
  • Zhengrong Liu

Abstract

We study the nonlinear waves described by Schamel‐Korteweg‐de Vries equation ut + (au1/2 + bu)ux + δuxxx = 0. Two new types of nonlinear waves called compacton‐like waves and kink‐like waves are displayed. Furthermore, two kinds of new bifurcation phenomena are revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which are the bell‐shape solitary waves, the blow‐up waves, the valley‐shape solitary waves, the kink‐like waves, and the compacton‐like waves. The second phenomenon is that the periodic‐blow‐up wave can be bifurcated from the smooth periodic wave.

Suggested Citation

  • Yun Wu & Zhengrong Liu, 2013. "New Types of Nonlinear Waves and Bifurcation Phenomena in Schamel‐Korteweg‐de Vries Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:483492
    DOI: 10.1155/2013/483492
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    References listed on IDEAS

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    1. Zhenshu Wen, 2012. "Extension on Bifurcations of Traveling Wave Solutions for a Two‐Component Fornberg‐Whitham Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Xianbin Wu & Weiguo Rui & Xiaochun Hong, 2013. "A Generalized KdV Equation of Neglecting the Highest-Order Infinitesimal Term and Its Exact Traveling Wave Solutions," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-15, March.
    3. Wazwaz, Abdul-Majid, 2006. "Two reliable methods for solving variants of the KdV equation with compact and noncompact structures," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 454-462.
    4. Ming Song, 2012. "Application of Bifurcation Method to the Generalized Zakharov Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-8, November.
    5. Xianbin Wu & Weiguo Rui & Xiaochun Hong, 2013. "A Generalized KdV Equation of Neglecting the Highest‐Order Infinitesimal Term and Its Exact Traveling Wave Solutions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    6. Ming Song, 2012. "Application of Bifurcation Method to the Generalized Zakharov Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    7. Zhenshu Wen, 2012. "Extension on Bifurcations of Traveling Wave Solutions for a Two-Component Fornberg-Whitham Equation," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-15, December.
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    Cited by:

    1. Xueqin Wang & Yadong Shang & Huahui Di, 2017. "Exact Solutions for the Wick‐Type Stochastic Schamel‐Korteweg‐de Vries Equation," Advances in Mathematical Physics, John Wiley & Sons, vol. 2017(1).
    2. Zhenshu Wen, 2014. "New Exact Explicit Nonlinear Wave Solutions for the Broer‐Kaup Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).

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