Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations
A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are pre...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-09-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/9/1547 |
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| author | Stephen C. Anco Bao Wang |
| author_facet | Stephen C. Anco Bao Wang |
| author_sort | Stephen C. Anco |
| collection | DOAJ |
| description | A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics. |
| first_indexed | 2024-03-10T16:12:41Z |
| format | Article |
| id | doaj.art-830133b79b6b48228756d879cafc634f |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T16:12:41Z |
| publishDate | 2020-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-830133b79b6b48228756d879cafc634f2023-11-20T14:18:48ZengMDPI AGSymmetry2073-89942020-09-01129154710.3390/sym12091547Geometrical Formulation for Adjoint-Symmetries of Partial Differential EquationsStephen C. Anco0Bao Wang1Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, CanadaDepartment of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, CanadaA geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.https://www.mdpi.com/2073-8994/12/9/1547adjoint-symmetryone-formsymmetryvector fieldgeometrical formulation |
| spellingShingle | Stephen C. Anco Bao Wang Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations Symmetry adjoint-symmetry one-form symmetry vector field geometrical formulation |
| title | Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations |
| title_full | Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations |
| title_fullStr | Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations |
| title_full_unstemmed | Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations |
| title_short | Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations |
| title_sort | geometrical formulation for adjoint symmetries of partial differential equations |
| topic | adjoint-symmetry one-form symmetry vector field geometrical formulation |
| url | https://www.mdpi.com/2073-8994/12/9/1547 |
| work_keys_str_mv | AT stephencanco geometricalformulationforadjointsymmetriesofpartialdifferentialequations AT baowang geometricalformulationforadjointsymmetriesofpartialdifferentialequations |