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Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

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  • Maria Santos Bruzón

    (Department of Mathematics, Faculty of Sciences, University of Cádiz, Puerto Real, 11510 Cádiz, Spain)

  • Gaetana Gambino

    (Department of Mathematics and Computer Science, University of Palermo, Via Archirafi, 9013 Palermo, Italy)

  • Maria Luz Gandarias

    (Department of Mathematics, Faculty of Sciences, University of Cádiz, Puerto Real, 11510 Cádiz, Spain)

Abstract

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively.

Suggested Citation

  • Maria Santos Bruzón & Gaetana Gambino & Maria Luz Gandarias, 2021. "Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions," Mathematics, MDPI, vol. 9(9), pages 1-20, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:1009-:d:546144
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    References listed on IDEAS

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    1. Russo, Matthew & Choudhury, S. Roy, 2017. "Analytic solutions of a microstructure PDE and the KdV and Kadomtsev–Petviashvili equations by invariant Painlevé analysis and generalized Hirota techniques," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 228-239.
    2. Wang, Xia, 2009. "Si’lnikov chaos and Hopf bifurcation analysis of Rucklidge system," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2208-2217.
    3. Kara, A.H. & Razborova, Polina & Biswas, Anjan, 2015. "Solitons and conservation laws of coupled Ostrovsky equation for internal waves," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 95-99.
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