Singularities of Type-Q ABS Equations
The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS) in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147397 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Singularities of Type-Q ABS Equations / J. Atkinson // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 10 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862581441187020800 |
|---|---|
| author | Atkinson, J. |
| author_facet | Atkinson, J. |
| citation_txt | Singularities of Type-Q ABS Equations / J. Atkinson // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 10 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS) in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the relationship between the singularities and the principal integrability feature of multidimensional consistency. These questions are considered in the global setting and therefore extend previous considerations of singularities which have been local. What emerges are some simple geometric criteria that determine the allowed singularities, and the interesting discovery that generically the presence of singularities leads to a breakdown in the global consistency of such systems despite their local consistency property. This failure to be globally consistent is quantified by introducing a natural notion of monodromy for isolated singularities.
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| first_indexed | 2025-11-26T22:57:53Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147397 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-26T22:57:53Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Atkinson, J. 2019-02-14T17:29:25Z 2019-02-14T17:29:25Z 2011 Singularities of Type-Q ABS Equations / J. Atkinson // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 10 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q58 DOI:10.3842/SIGMA.2011.073 https://nasplib.isofts.kiev.ua/handle/123456789/147397 The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS) in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the relationship between the singularities and the principal integrability feature of multidimensional consistency. These questions are considered in the global setting and therefore extend previous considerations of singularities which have been local. What emerges are some simple geometric criteria that determine the allowed singularities, and the interesting discovery that generically the presence of singularities leads to a breakdown in the global consistency of such systems despite their local consistency property. This failure to be globally consistent is quantified by introducing a natural notion of monodromy for isolated singularities. This paper is a contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)” (June 14–18, 2010, Varna, Bulgaria). The full collection is available at http://www.emis.de/journals/SIGMA/SIDE-9.html.
 The author gratefully acknowledges helpful discussions with Nalini Joshi. The research was funded by Australian Research Council Discovery Grants DP 0985615 and DP 110104151. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Singularities of Type-Q ABS Equations Article published earlier |
| spellingShingle | Singularities of Type-Q ABS Equations Atkinson, J. |
| title | Singularities of Type-Q ABS Equations |
| title_full | Singularities of Type-Q ABS Equations |
| title_fullStr | Singularities of Type-Q ABS Equations |
| title_full_unstemmed | Singularities of Type-Q ABS Equations |
| title_short | Singularities of Type-Q ABS Equations |
| title_sort | singularities of type-q abs equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147397 |
| work_keys_str_mv | AT atkinsonj singularitiesoftypeqabsequations |