The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle i...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2017
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148568 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs / B. Güneysu, M.J. Pflaum // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 50 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862669788755525632 |
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| author | Güneysu, B. Pflaum, M.J. |
| author_facet | Güneysu, B. Pflaum, M.J. |
| citation_txt | The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs / B. Güneysu, M.J. Pflaum // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 50 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.
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| first_indexed | 2025-12-07T15:28:47Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148568 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:28:47Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
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| spelling | Güneysu, B. Pflaum, M.J. 2019-02-18T15:58:27Z 2019-02-18T15:58:27Z 2017 The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs / B. Güneysu, M.J. Pflaum // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 50 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58A05; 58A20; 35A30 DOI:10.3842/SIGMA.2017.003 https://nasplib.isofts.kiev.ua/handle/123456789/148568 In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable. The first named author (B.G.) is indebted to W.M. Seiler for many discussions on jet bundles,
 and would also like to thank B. Kruglikov and A.D. Lewis for helpful discussions. B.G. has
 been financially supported by the SFB 647: Raum–Zeit–Materie, and would like to thank the
 University of Colorado at Boulder for its hospitality. The second named author (M.P.) has been
 partially supported by NSF grant DMS 1105670 and by a Simons Foundation collaboration
 grant, award nr. 359389. M.P. would also like to thank Humboldt-University, Berlin and the
 Max-Planck-Institute for Mathematics of the Sciences, Leipzig for their hospitality. Last but not
 least the authors thank the anonymous referees for constructive advice which helped to improve
 the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs Article published earlier |
| spellingShingle | The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs Güneysu, B. Pflaum, M.J. |
| title | The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs |
| title_full | The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs |
| title_fullStr | The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs |
| title_full_unstemmed | The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs |
| title_short | The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs |
| title_sort | profinite dimensional manifold structure of formal solution spaces of formally integrable pdes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148568 |
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