A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149230 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | A Quasi-Lie Schemes Approach to Second-Order Gambier Equations / J.F. Cariñena, P. Guha, L. de Lucas // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 56 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862538877252665344 |
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| author | Cariñena, J.F. Guha, P. de Lucas, J. |
| author_facet | Cariñena, J.F. Guha, P. de Lucas, J. |
| citation_txt | A Quasi-Lie Schemes Approach to Second-Order Gambier Equations / J.F. Cariñena, P. Guha, L. de Lucas // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 56 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators.
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| first_indexed | 2025-11-24T15:04:59Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149230 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T15:04:59Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
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| spelling | Cariñena, J.F. Guha, P. de Lucas, J. 2019-02-19T19:03:08Z 2019-02-19T19:03:08Z 2013 A Quasi-Lie Schemes Approach to Second-Order Gambier Equations / J.F. Cariñena, P. Guha, L. de Lucas // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 56 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34A26; 34A05; 34A34; 17B66; 53Z05 DOI: http://dx.doi.org/10.3842/SIGMA.2013.026 https://nasplib.isofts.kiev.ua/handle/123456789/149230 A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants
 and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
 The research of J.F. Cari˜nena and J. de Lucas was supported by the Polish National Science
 Centre under the grant HARMONIA Nr 2012/04/M/ST1/00523. They also acknowledge partial
 financial support by research projects MTM–2009–11154 (MEC) and E24/1 (DGA). J. de Lucas
 would like to thank for a research grant FMI40/10 (DGA) to accomplish a research stay in the
 University of Zaragoza. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Quasi-Lie Schemes Approach to Second-Order Gambier Equations Article published earlier |
| spellingShingle | A Quasi-Lie Schemes Approach to Second-Order Gambier Equations Cariñena, J.F. Guha, P. de Lucas, J. |
| title | A Quasi-Lie Schemes Approach to Second-Order Gambier Equations |
| title_full | A Quasi-Lie Schemes Approach to Second-Order Gambier Equations |
| title_fullStr | A Quasi-Lie Schemes Approach to Second-Order Gambier Equations |
| title_full_unstemmed | A Quasi-Lie Schemes Approach to Second-Order Gambier Equations |
| title_short | A Quasi-Lie Schemes Approach to Second-Order Gambier Equations |
| title_sort | quasi-lie schemes approach to second-order gambier equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149230 |
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