Invariant Reduction for Partial Differential Equations. II: The General Framework
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917357144768512 |
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| author | Druzhkov, Kostya Cheviakov, Alexei |
| author_facet | Druzhkov, Kostya Cheviakov, Alexei |
| contents | For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $φ$, invariant solutions satisfy the reduced system $F = φ= 0$. We propose a framework that allows, for every $X$-invariant conservation law, presymplectic structure, variational principle, or another geometric structure of the given PDE system $F = 0$, to systematically calculate its corresponding reduced form that describes the corresponding structure for the reduced system $F = φ= 0$. In particular, we show in what way Noether's theorem holding for the given PDE system is inherited by the reduced PDE system. We consider several detailed examples, including cases of point and higher symmetry invariance. The proposed framework is directly applicable to a wide range of PDE models, including complex PDE systems of contemporary interest arising across disciplines, where symmetry reduction is essential for analysis and simulation, as well as to integrable, Lagrangian, and gauge systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariant Reduction for Partial Differential Equations. II: The General Framework Druzhkov, Kostya Cheviakov, Alexei Exactly Solvable and Integrable Systems Mathematical Physics Differential Geometry For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $φ$, invariant solutions satisfy the reduced system $F = φ= 0$. We propose a framework that allows, for every $X$-invariant conservation law, presymplectic structure, variational principle, or another geometric structure of the given PDE system $F = 0$, to systematically calculate its corresponding reduced form that describes the corresponding structure for the reduced system $F = φ= 0$. In particular, we show in what way Noether's theorem holding for the given PDE system is inherited by the reduced PDE system. We consider several detailed examples, including cases of point and higher symmetry invariance. The proposed framework is directly applicable to a wide range of PDE models, including complex PDE systems of contemporary interest arising across disciplines, where symmetry reduction is essential for analysis and simulation, as well as to integrable, Lagrangian, and gauge systems. |
| title | Invariant Reduction for Partial Differential Equations. II: The General Framework |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2501.09313 |