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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2014 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2014
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146625 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862533003876499456 |
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| author | Startsev, S.Y. |
| author_facet | Startsev, S.Y. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-11-24T06:30:35Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146625 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T06:30:35Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
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| spelling | Startsev, S.Y. 2019-02-10T10:33:27Z 2019-02-10T10:33:27Z 2014 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 https://nasplib.isofts.kiev.ua/handle/123456789/146625 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. The author thanks the referees for useful suggestions. This work is partially supported by the
 Russian Foundation for Basic Research (grant number 13-01-00070-a). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Non-Point Invertible Transformations and Integrability of Partial Difference Equations Article published earlier |
| spellingShingle | Non-Point Invertible Transformations and Integrability of Partial Difference Equations Startsev, S.Y. |
| title | 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_short | Non-Point Invertible Transformations and Integrability of Partial Difference Equations |
| title_sort | non-point invertible transformations and integrability of partial difference equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146625 |
| work_keys_str_mv | AT startsevsy nonpointinvertibletransformationsandintegrabilityofpartialdifferenceequations |