Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions
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 met...
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MDPI AG
2021-04-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/9/1009 |
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| author | Maria Santos Bruzón Gaetana Gambino Maria Luz Gandarias |
| author_facet | Maria Santos Bruzón Gaetana Gambino Maria Luz Gandarias |
| author_sort | Maria Santos Bruzón |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T04:56:40Z |
| format | Article |
| id | doaj.art-e7be787272f549f8b2bd677119ac243b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T04:56:40Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-e7be787272f549f8b2bd677119ac243b2023-12-03T13:04:54ZengMDPI AGMathematics2227-73902021-04-0199100910.3390/math9091009Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front SolutionsMaria Santos Bruzón0Gaetana Gambino1Maria Luz Gandarias2Department of Mathematics, Faculty of Sciences, University of Cádiz, Puerto Real, 11510 Cádiz, SpainDepartment of Mathematics and Computer Science, University of Palermo, Via Archirafi, 9013 Palermo, ItalyDepartment of Mathematics, Faculty of Sciences, University of Cádiz, Puerto Real, 11510 Cádiz, SpainIn 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.https://www.mdpi.com/2227-7390/9/9/1009generalized Camassa–Holm equationsnonclassical symmetriesmultiplier methodconservation lawsdouble reductionhomoclinic and heteroclinic orbits |
| spellingShingle | Maria Santos Bruzón Gaetana Gambino Maria Luz Gandarias Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions Mathematics generalized Camassa–Holm equations nonclassical symmetries multiplier method conservation laws double reduction homoclinic and heteroclinic orbits |
| title | Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions |
| title_full | Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions |
| title_fullStr | Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions |
| title_full_unstemmed | Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions |
| title_short | Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions |
| title_sort | generalized camassa holm equations symmetry conservation laws and regular pulse and front solutions |
| topic | generalized Camassa–Holm equations nonclassical symmetries multiplier method conservation laws double reduction homoclinic and heteroclinic orbits |
| url | https://www.mdpi.com/2227-7390/9/9/1009 |
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