Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type
The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evol...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2013 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2013
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/149228 |
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| Zitieren: | Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862643835366014976 |
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| author | Vassiliou, P.J. |
| author_facet | Vassiliou, P.J. |
| citation_txt | Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value problems. Additionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed.
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| first_indexed | 2025-12-01T08:13:07Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-149228 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T08:13:07Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Vassiliou, P.J. 2019-02-19T19:02:16Z 2019-02-19T19:02:16Z 2013 Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A35; 53A55; 58A15; 58A20; 58A30 DOI: http://dx.doi.org/10.3842/SIGMA.2013.024 https://nasplib.isofts.kiev.ua/handle/123456789/149228 The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value problems. Additionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed. 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.
 I’m grateful to the three anonymous referees for their close reading of the manuscript and for
 making suggestions which considerably improved the paper. I would like to acknowledge, with
 my thanks, the early involvement of Jordane Math´e for carefully working together through
 the calculations in Section 4 which formed a portion of his internship from the Ecole normale
 sup´erieure de Cachan, France. Much of the research for this paper was carried out while I was
 a Visiting Fellow at the Mathematical Sciences Institute of the Australian National University,
 Canberra. The hospitality of the MSI is gratefully acknowledged. In particular, I thank Mike
 Eastwood and the Dif ferential Geometry Group for stimulating discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type Article published earlier |
| spellingShingle | Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type Vassiliou, P.J. |
| title | Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type |
| title_full | Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type |
| title_fullStr | Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type |
| title_full_unstemmed | Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type |
| title_short | Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type |
| title_sort | cauchy problem for a darboux integrable wave map system and equations of lie type |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149228 |
| work_keys_str_mv | AT vassilioupj cauchyproblemforadarbouxintegrablewavemapsystemandequationsoflietype |