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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| Дата: | 2013 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2013
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149228 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1492282019-02-20T01:23:09Z Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type Vassiliou, P.J. 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. 2013 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149228 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Vassiliou, P.J. |
| spellingShingle |
Vassiliou, P.J. Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Vassiliou, P.J. |
| author_sort |
Vassiliou, P.J. |
| title |
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_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_sort |
cauchy problem for a darboux integrable wave map system and equations of lie type |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149228 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT vassilioupj cauchyproblemforadarbouxintegrablewavemapsystemandequationsoflietype |
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2023-05-20T17:32:17Z |
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2023-05-20T17:32:17Z |
| _version_ |
1796153515873140736 |