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A new integrable equation with no smooth solitons

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  • Qiao, Zhijun
  • Liu, Liping

Abstract

In this paper, we propose a new completely integrable equation:mt=121m2xxx-121m2x,which has no smooth solitons. This equation is shown to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation. Studying this new equation, we develop two new kinds of soliton solutions under the inhomogeneous boundary condition lim|x|→∞m=B where B is nonzero constant. One is continuous and piecewise smooth “W/M”-shape-peaks solitary solution and the other one-single-peak soliton. The two new kinds of peaked solitons can not be written as the regular type peakon: ce-|x-ct|, where c is a constant. We will provide graphs to show those new kinds of peaked solitons.

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  • Qiao, Zhijun & Liu, Liping, 2009. "A new integrable equation with no smooth solitons," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 587-593.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:2:p:587-593
    DOI: 10.1016/j.chaos.2007.11.034
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    References listed on IDEAS

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    1. Parkes, E.J. & Vakhnenko, V.O., 2005. "Explicit solutions of the Camassa–Holm equation," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1309-1316.
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    Cited by:

    1. María S. Bruzón & Rafael de la Rosa & María L. Gandarias & Rita Tracinà, 2022. "Applications of Solvable Lie Algebras to a Class of Third Order Equations," Mathematics, MDPI, vol. 10(2), pages 1-19, January.
    2. María S. Bruzón & Rafael de la Rosa & María L. Gandarias & Rita Tracinà, 2022. "Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method," Mathematics, MDPI, vol. 10(6), pages 1-13, March.
    3. Hengtai Wang & Zhiwei Zou & Xin Shen, 2021. "Lie Symmetry Analysis, Self-Adjointness and Conservation Law for a Type of Nonlinear Equation," Mathematics, MDPI, vol. 9(12), pages 1-10, June.

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