C-Integrability Test for Discrete Equations via Multiple Scale Expansions
In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2010 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2010
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146506 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862709356111331328 |
|---|---|
| author | Scimiterna, C. Levi, D. |
| author_facet | Scimiterna, C. Levi, D. |
| citation_txt | C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential-difference equation. In this case, the equation satisfies the A₁, A₂ and A₃ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.
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| first_indexed | 2025-12-07T17:17:38Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146506 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:17:38Z |
| publishDate | 2010 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Scimiterna, C. Levi, D. 2019-02-09T19:34:04Z 2019-02-09T19:34:04Z 2010 C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34K99; 34E13; 37K10; 37J30 DOI:10.3842/SIGMA.2010.070 https://nasplib.isofts.kiev.ua/handle/123456789/146506 In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential-difference equation. In this case, the equation satisfies the A₁, A₂ and A₃ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable. The authors have been partly supported by the Italian Ministry of Education and Research, PRIN “Nonlinear waves: integrable finite dimensional reductions and discretizations” from 2007 to 2009 and PRIN “Continuous and discrete nonlinear integrable evolutions: from water waves to symplectic maps” from 2010. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications C-Integrability Test for Discrete Equations via Multiple Scale Expansions Article published earlier |
| spellingShingle | C-Integrability Test for Discrete Equations via Multiple Scale Expansions Scimiterna, C. Levi, D. |
| title | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_full | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_fullStr | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_full_unstemmed | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_short | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_sort | c-integrability test for discrete equations via multiple scale expansions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146506 |
| work_keys_str_mv | AT scimiternac cintegrabilitytestfordiscreteequationsviamultiplescaleexpansions AT levid cintegrabilitytestfordiscreteequationsviamultiplescaleexpansions |