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

Maria Santos Bruzón, Gaetana Gambino and Maria Luz Gandarias
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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

Mathematics, 2021, vol. 9, issue 9, 1-20

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.

Keywords: generalized Camassa–Holm equations; nonclassical symmetries; multiplier method; conservation laws; double reduction; homoclinic and heteroclinic orbits; multi-infinite series solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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