Painlevé Analysis and Similarity Reductions for the Magma Equation
In this paper, we examine a generalized magma equation for rational values of two parameters, m and n. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlevé ODE test is then applied to the travelling wave reduction, and the pairs of m...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2006 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2006
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146099 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Painlevé Analysis and Similarity Reductions for the Magma Equation / S.E. Harris, P.A. Clarkson // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 34 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862601608359051264 |
|---|---|
| author | Harris, S.E. Clarkson, P.A. |
| author_facet | Harris, S.E. Clarkson, P.A. |
| citation_txt | Painlevé Analysis and Similarity Reductions for the Magma Equation / S.E. Harris, P.A. Clarkson // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 34 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we examine a generalized magma equation for rational values of two parameters, m and n. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlevé ODE test is then applied to the travelling wave reduction, and the pairs of m and n which pass the test are identified. These particular pairs are further subjected to the ODE test on their other symmetry reductions. Only two cases remain which pass the ODE test for all such symmetry reductions and these are completely integrable. The case when m = 0, n = −1 is related to the Hirota-Satsuma equation and for m = ½, n = −½, it is a real, generalized, pumped Maxwell-Bloch equation.
|
| first_indexed | 2025-11-28T02:24:37Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146099 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-28T02:24:37Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Harris, S.E. Clarkson, P.A. 2019-02-07T13:13:12Z 2019-02-07T13:13:12Z 2006 Painlevé Analysis and Similarity Reductions for the Magma Equation / S.E. Harris, P.A. Clarkson // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 34 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 35C05; 35Q58; 37K10 https://nasplib.isofts.kiev.ua/handle/123456789/146099 In this paper, we examine a generalized magma equation for rational values of two parameters, m and n. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlevé ODE test is then applied to the travelling wave reduction, and the pairs of m and n which pass the test are identified. These particular pairs are further subjected to the ODE test on their other symmetry reductions. Only two cases remain which pass the ODE test for all such symmetry reductions and these are completely integrable. The case when m = 0, n = −1 is related to the Hirota-Satsuma equation and for m = ½, n = −½, it is a real, generalized, pumped Maxwell-Bloch equation. It is a pleasure to thank Elizabeth Mansfield and Andrew Pickering for their helpful comments and discussions. SEH gratefully acknowledges the support of a Darby Fellowship at Lincoln College, Oxford and a Grace Chisholm Young Fellowship from the London Mathematical Society for the period in which this research was carried out. PAC thanks the Isaac Newton Institute, Cambridge for their hospitality during his visit as part of the programme on “Painlev´e Equations and Monodromy Problems” when this paper was completed. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Painlevé Analysis and Similarity Reductions for the Magma Equation Article published earlier |
| spellingShingle | Painlevé Analysis and Similarity Reductions for the Magma Equation Harris, S.E. Clarkson, P.A. |
| title | Painlevé Analysis and Similarity Reductions for the Magma Equation |
| title_full | Painlevé Analysis and Similarity Reductions for the Magma Equation |
| title_fullStr | Painlevé Analysis and Similarity Reductions for the Magma Equation |
| title_full_unstemmed | Painlevé Analysis and Similarity Reductions for the Magma Equation |
| title_short | Painlevé Analysis and Similarity Reductions for the Magma Equation |
| title_sort | painlevé analysis and similarity reductions for the magma equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146099 |
| work_keys_str_mv | AT harrisse painleveanalysisandsimilarityreductionsforthemagmaequation AT clarksonpa painleveanalysisandsimilarityreductionsforthemagmaequation |