On the Limited Role of Conservation Laws in the Double Reduction Routine for Partial Differential Equations
- Sinkala, Winter, Kakuli, Charles Molahlehli
- Authors: Sinkala, Winter , Kakuli, Charles Molahlehli
- Date: 2024
- Subjects: Double reduction , Lie symmetry analysis , Invariant solution
- Language: English
- Type: Article
- Identifier: http://hdl.handle.net/11260/14892 , vital:79859 , DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5318
- Description: The double reduction method for finding invariant solutions of a given partial differential equation (PDE) provides for the reduction of a q-th order PDE that admits a Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order q-1.In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of one less than that of the PDE. We have outlined the steps involved in this variation of the double reduction method and illustrated the routine using five PDEs.
- Full Text:
- Date Issued: 2024
- Authors: Sinkala, Winter , Kakuli, Charles Molahlehli
- Date: 2024
- Subjects: Double reduction , Lie symmetry analysis , Invariant solution
- Language: English
- Type: Article
- Identifier: http://hdl.handle.net/11260/14892 , vital:79859 , DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5318
- Description: The double reduction method for finding invariant solutions of a given partial differential equation (PDE) provides for the reduction of a q-th order PDE that admits a Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order q-1.In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of one less than that of the PDE. We have outlined the steps involved in this variation of the double reduction method and illustrated the routine using five PDEs.
- Full Text:
- Date Issued: 2024
Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method
- Kakuli, Charles Molahlehi, Sinkala, Winter, Masemola, Phetogo
- Authors: Kakuli, Charles Molahlehi , Sinkala, Winter , Masemola, Phetogo
- Date: 2023
- Subjects: Double reaction , Hunter-Saxton equation , Lie symmetry analysis , Conservation law , Invariant solution
- Language: English
- Type: Article
- Identifier: http://hdl.handle.net/11260/14911 , vital:79853 , DOI: https://doi.org/10.3390/mca28050092
- 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 ismore 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.
- Full Text:
- Date Issued: 2023
- Authors: Kakuli, Charles Molahlehi , Sinkala, Winter , Masemola, Phetogo
- Date: 2023
- Subjects: Double reaction , Hunter-Saxton equation , Lie symmetry analysis , Conservation law , Invariant solution
- Language: English
- Type: Article
- Identifier: http://hdl.handle.net/11260/14911 , vital:79853 , DOI: https://doi.org/10.3390/mca28050092
- 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 ismore 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.
- Full Text:
- Date Issued: 2023
- «
- ‹
- 1
- ›
- »