Recursions of Symmetry Orbits and Reduction without Reduction
We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Ampère equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie...
Збережено в:
| Дата: | 2011 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2011
|
| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146860 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Recursions of Symmetry Orbits and Reduction without Reduction / M.B. Sheftel, A.A. Malykh // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 14 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-146860 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1468602020-10-13T23:59:18Z Recursions of Symmetry Orbits and Reduction without Reduction Malykh, A.A. Sheftel, M.B. We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Ampère equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat Kähler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation. 2011 Article Recursions of Symmetry Orbits and Reduction without Reduction / M.B. Sheftel, A.A. Malykh // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q75; 83C15 DOI:10.3842/SIGMA.2011.043 http://dspace.nbuv.gov.ua/handle/123456789/146860 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 |
We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Ampère equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat Kähler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation. |
| format |
Article |
| author |
Malykh, A.A. Sheftel, M.B. |
| spellingShingle |
Malykh, A.A. Sheftel, M.B. Recursions of Symmetry Orbits and Reduction without Reduction Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Malykh, A.A. Sheftel, M.B. |
| author_sort |
Malykh, A.A. |
| title |
Recursions of Symmetry Orbits and Reduction without Reduction |
| title_short |
Recursions of Symmetry Orbits and Reduction without Reduction |
| title_full |
Recursions of Symmetry Orbits and Reduction without Reduction |
| title_fullStr |
Recursions of Symmetry Orbits and Reduction without Reduction |
| title_full_unstemmed |
Recursions of Symmetry Orbits and Reduction without Reduction |
| title_sort |
recursions of symmetry orbits and reduction without reduction |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146860 |
| citation_txt |
Recursions of Symmetry Orbits and Reduction without Reduction / M.B. Sheftel, A.A. Malykh // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 14 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT malykhaa recursionsofsymmetryorbitsandreductionwithoutreduction AT sheftelmb recursionsofsymmetryorbitsandreductionwithoutreduction |
| first_indexed |
2023-10-18T21:36:35Z |
| last_indexed |
2023-10-18T21:36:35Z |
| _version_ |
1796153278731386880 |