Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method
This study investigates via Lie symmetry analysis the Hunter–Saxton equation, an equation relevant to the theoretical analysis of nematic liquid crystals. We employ the multiplier method to obtain conservation laws of the equation that arise from first-order multipliers. Conservation laws of the equ...
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MDPI AG
2023-08-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/28/5/92 |
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| author | Molahlehi Charles Kakuli Winter Sinkala Phetogo Masemola |
| author_facet | Molahlehi Charles Kakuli Winter Sinkala Phetogo Masemola |
| author_sort | Molahlehi Charles Kakuli |
| collection | DOAJ |
| description | This study investigates via Lie symmetry analysis the Hunter–Saxton equation, an equation relevant to the theoretical analysis of nematic liquid crystals. We employ the multiplier method to obtain conservation laws of the equation that arise from first-order multipliers. Conservation laws of the equation, combined with the admitted Lie point symmetries, enable us to perform symmetry reductions by employing the double reduction method. The method exploits the relationship between symmetries and conservation laws to reduce both the number of variables and the order of the equation. Five nontrivial conservation laws of the Hunter–Saxton equation are derived, four of which are found to have associated Lie point symmetries. Applying the double reduction method to the equation results in a set of first-order ordinary differential equations, the solutions of which represent invariant solutions for the equation. While the double reduction method may be more complex to implement than the classical method, since it involves finding Lie point symmetries and deriving conservation laws, it has some advantages over the classical method of reducing PDEs. Firstly, it is more efficient in that it can reduce the number of variables and order of the equation in a single step. Secondly, by incorporating conservation laws, physically meaningful solutions that satisfy important physical constraints can be obtained. |
| first_indexed | 2024-03-10T21:04:35Z |
| format | Article |
| id | doaj.art-4567141d26da455daa8d7d86d7838cc3 |
| institution | Directory Open Access Journal |
| issn | 1300-686X 2297-8747 |
| language | English |
| last_indexed | 2024-03-10T21:04:35Z |
| publishDate | 2023-08-01 |
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| series | Mathematical and Computational Applications |
| spelling | doaj.art-4567141d26da455daa8d7d86d7838cc32023-11-19T17:15:35ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472023-08-012859210.3390/mca28050092Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction MethodMolahlehi Charles Kakuli0Winter Sinkala1Phetogo Masemola2Department of Mathematical Sciences and Computing, Faculty of Natural Sciences, Walter Sisulu University, Private Bag X1, Mthatha 5117, South AfricaDepartment of Mathematical Sciences and Computing, Faculty of Natural Sciences, Walter Sisulu University, Private Bag X1, Mthatha 5117, South AfricaSchool of Mathematics, University of the Witwatersrand, Johannesburg 2000, South AfricaThis study investigates via Lie symmetry analysis the Hunter–Saxton equation, an equation relevant to the theoretical analysis of nematic liquid crystals. We employ the multiplier method to obtain conservation laws of the equation that arise from first-order multipliers. Conservation laws of the equation, combined with the admitted Lie point symmetries, enable us to perform symmetry reductions by employing the double reduction method. The method exploits the relationship between symmetries and conservation laws to reduce both the number of variables and the order of the equation. Five nontrivial conservation laws of the Hunter–Saxton equation are derived, four of which are found to have associated Lie point symmetries. Applying the double reduction method to the equation results in a set of first-order ordinary differential equations, the solutions of which represent invariant solutions for the equation. While the double reduction method may be more complex to implement than the classical method, since it involves finding Lie point symmetries and deriving conservation laws, it has some advantages over the classical method of reducing PDEs. Firstly, it is more efficient in that it can reduce the number of variables and order of the equation in a single step. Secondly, by incorporating conservation laws, physically meaningful solutions that satisfy important physical constraints can be obtained.https://www.mdpi.com/2297-8747/28/5/92double reductionHunter–Saxton equationlie symmetry analysisconservation lawinvariant solution |
| spellingShingle | Molahlehi Charles Kakuli Winter Sinkala Phetogo Masemola Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method Mathematical and Computational Applications double reduction Hunter–Saxton equation lie symmetry analysis conservation law invariant solution |
| title | Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method |
| title_full | Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method |
| title_fullStr | Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method |
| title_full_unstemmed | Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method |
| title_short | Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method |
| title_sort | conservation laws and symmetry reductions of the hunter saxton equation via the double reduction method |
| topic | double reduction Hunter–Saxton equation lie symmetry analysis conservation law invariant solution |
| url | https://www.mdpi.com/2297-8747/28/5/92 |
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