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Article

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

1
Department of Mathematics, Faculty of Sciences, University of Cádiz, Puerto Real, 11510 Cádiz, Spain
2
Department of Mathematics and Computer Science, University of Palermo, Via Archirafi, 9013 Palermo, Italy
*
Author to whom correspondence should be addressed.
Academic Editor: Andrei Dmitrievich Polyanin
Mathematics 2021, 9(9), 1009; https://doi.org/10.3390/math9091009
Received: 31 March 2021 / Revised: 24 April 2021 / Accepted: 27 April 2021 / Published: 29 April 2021
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. View Full-Text
Keywords: generalized Camassa–Holm equations; nonclassical symmetries; multiplier method; conservation laws; double reduction; homoclinic and heteroclinic orbits; multi-infinite series solutions generalized Camassa–Holm equations; nonclassical symmetries; multiplier method; conservation laws; double reduction; homoclinic and heteroclinic orbits; multi-infinite series solutions
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MDPI and ACS Style

Bruzón, M.S.; Gambino, G.; Gandarias, M.L. Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions. Mathematics 2021, 9, 1009. https://doi.org/10.3390/math9091009

AMA Style

Bruzón MS, Gambino G, Gandarias ML. Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions. Mathematics. 2021; 9(9):1009. https://doi.org/10.3390/math9091009

Chicago/Turabian Style

Bruzón, Maria S., Gaetana Gambino, and Maria L. Gandarias 2021. "Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions" Mathematics 9, no. 9: 1009. https://doi.org/10.3390/math9091009

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