Non-Point Invertible Transformations and Integrability of Partial Difference Equations
This article is devoted to the partial difference quad-graph equations that can be represented in the form φ(u(i+1,j),u(i+1,j+1))=ψ(u(i,j),u(i,j+1)), where the map (w,z)→(φ(w,z),ψ(w,z)) is injective. The transformation v(i,j)=φ(u(i,j),u(i,j+1)) relates any of such equations to a quad-graph equation....
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| Дата: | 2014 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2014
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146625 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Non-Point Invertible Transformations and Integrability of Partial Difference Equations / S.Y. Startsev // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1466252020-10-17T00:33:02Z Non-Point Invertible Transformations and Integrability of Partial Difference Equations Startsev, S.Y. This article is devoted to the partial difference quad-graph equations that can be represented in the form φ(u(i+1,j),u(i+1,j+1))=ψ(u(i,j),u(i,j+1)), where the map (w,z)→(φ(w,z),ψ(w,z)) is injective. The transformation v(i,j)=φ(u(i,j),u(i,j+1)) relates any of such equations to a quad-graph equation. It is proved that this transformation maps Darboux integrable equations of the above form into Darboux integrable equations again and decreases the orders of the transformed integrals by one in the j-direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the j-direction are described under an additional assumption. The transformation also maps symmetries of the original equations into symmetries of the transformed equations (i.e. preserves the integrability in the sense of the symmetry approach) and acts as a difference substitution for symmetries of a special form. The latter fact allows us to derive necessary conditions of Darboux integrability for the equations defined in the first sentence of the abstract. 2014 Article Non-Point Invertible Transformations and Integrability of Partial Difference Equations / S.Y. Startsev // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 39A14; 37K05; 37K10; 37K35 DOI:10.3842/SIGMA.2014.066 http://dspace.nbuv.gov.ua/handle/123456789/146625 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 |
This article is devoted to the partial difference quad-graph equations that can be represented in the form φ(u(i+1,j),u(i+1,j+1))=ψ(u(i,j),u(i,j+1)), where the map (w,z)→(φ(w,z),ψ(w,z)) is injective. The transformation v(i,j)=φ(u(i,j),u(i,j+1)) relates any of such equations to a quad-graph equation. It is proved that this transformation maps Darboux integrable equations of the above form into Darboux integrable equations again and decreases the orders of the transformed integrals by one in the j-direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the j-direction are described under an additional assumption. The transformation also maps symmetries of the original equations into symmetries of the transformed equations (i.e. preserves the integrability in the sense of the symmetry approach) and acts as a difference substitution for symmetries of a special form. The latter fact allows us to derive necessary conditions of Darboux integrability for the equations defined in the first sentence of the abstract. |
| format |
Article |
| author |
Startsev, S.Y. |
| spellingShingle |
Startsev, S.Y. Non-Point Invertible Transformations and Integrability of Partial Difference Equations Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Startsev, S.Y. |
| author_sort |
Startsev, S.Y. |
| title |
Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
| title_short |
Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
| title_full |
Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
| title_fullStr |
Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
| title_full_unstemmed |
Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
| title_sort |
non-point invertible transformations and integrability of partial difference equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146625 |
| citation_txt |
Non-Point Invertible Transformations and Integrability of Partial Difference Equations / S.Y. Startsev // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 18 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT startsevsy nonpointinvertibletransformationsandintegrabilityofpartialdifferenceequations |
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2025-07-11T00:20:52Z |
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2025-07-11T00:20:52Z |
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