Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory
Darboux developed an ingenious algebraic mechanism to construct infinite chains of ''integrable'' second-order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance,...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211868 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory. Primitivo Acosta-Humánez, Moulay Barkatou, Raquel Sánchez-Cauce and Jacques-Arthur Weil. SIGMA 19 (2023), 016, 29 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862748303003746304 |
|---|---|
| author | Acosta-Humánez, Primitivo Barkatou, Moulay Sánchez-Cauce, Raquel Weil, Jacques-Arthur |
| author_facet | Acosta-Humánez, Primitivo Barkatou, Moulay Sánchez-Cauce, Raquel Weil, Jacques-Arthur |
| citation_txt | Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory. Primitivo Acosta-Humánez, Moulay Barkatou, Raquel Sánchez-Cauce and Jacques-Arthur Weil. SIGMA 19 (2023), 016, 29 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Darboux developed an ingenious algebraic mechanism to construct infinite chains of ''integrable'' second-order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance, in quantum mechanics (where they provide useful tools for supersymmetric quantum mechanics), in soliton theory, Lax pairs, and many other fields involving hierarchies of equations. In this paper, we propose a method that allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third-order orthogonal systems ((3, ) systems) as well as a framework to extend Darboux transformations to any symmetric power of SL(2, ℂ)-systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials. All results in this paper have been implemented and tested in the computer algebra system Maple.
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| first_indexed | 2026-04-17T19:46:54Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211868 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T19:46:54Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Acosta-Humánez, Primitivo Barkatou, Moulay Sánchez-Cauce, Raquel Weil, Jacques-Arthur 2026-01-14T09:53:32Z 2023 Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory. Primitivo Acosta-Humánez, Moulay Barkatou, Raquel Sánchez-Cauce and Jacques-Arthur Weil. SIGMA 19 (2023), 016, 29 pages 1815-0659 2020 Mathematics Subject Classification: 12H05; 35Q40; 81Q60 arXiv:2101.07470 https://nasplib.isofts.kiev.ua/handle/123456789/211868 https://doi.org/10.3842/SIGMA.2023.016 Darboux developed an ingenious algebraic mechanism to construct infinite chains of ''integrable'' second-order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance, in quantum mechanics (where they provide useful tools for supersymmetric quantum mechanics), in soliton theory, Lax pairs, and many other fields involving hierarchies of equations. In this paper, we propose a method that allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third-order orthogonal systems ((3, ) systems) as well as a framework to extend Darboux transformations to any symmetric power of SL(2, ℂ)-systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials. All results in this paper have been implemented and tested in the computer algebra system Maple. The first author thanks the hospitality of XLim and the suggestions of J.J. Morales-Ruiz during the initial stage of this work. He was supported in the final stage of this paper by the FONDOCYT grants 2022-1D2-90 and 2022-1D2-091 from the Dominican Government (MESCYT). The third author thanks the Autonomous University of Madrid for the financial support for a research stay at XLim, where she started to work on this article. She also thanks the hospitality of XLim and the support of J.J. Morales-Ruiz to participate in this work. This work was partially supported by the grant TIN2016-77206-R from the Spanish Government, co-financed by the European Regional Development Fund. The third author received a postdoctoral grant (PEJD-2018-POST/TIC-9490) from Universidad Nacional de Educación a Distancia (UNED), co-financed by the Regional Government of Madrid and the Youth Employment Initiative (YEI) of the European Social Fund. The authors gratefully acknowledge the referees for their helpful comments and further references, which resulted in an improvement of the preliminary manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory Article published earlier |
| spellingShingle | Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory Acosta-Humánez, Primitivo Barkatou, Moulay Sánchez-Cauce, Raquel Weil, Jacques-Arthur |
| title | Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory |
| title_full | Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory |
| title_fullStr | Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory |
| title_full_unstemmed | Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory |
| title_short | Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory |
| title_sort | darboux transformations for orthogonal differential systems and differential galois theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211868 |
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