The classical M. A. Buhl problem, its Pfeiffer – Sato solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly type nonlinear equations
The survey is devoted to old and recent investigations of the classical M. A. Buhl problem of description of the compatible linear vector field equations and their general M. G. Pfeiffer and modern Lax – Sato-type special solutions. In particular, we analyze the related Lie-algebraic structures and...
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| Datum: | 2017 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
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Institute of Mathematics, NAS of Ukraine
2017
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1811 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507674472349696 |
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| author | Prykarpatsky, Ya. A. Samoilenko, A. M. Прикарпатський, Я. А. Самойленко, А. М. |
| author_facet | Prykarpatsky, Ya. A. Samoilenko, A. M. Прикарпатський, Я. А. Самойленко, А. М. |
| author_sort | Prykarpatsky, Ya. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:28:05Z |
| description | The survey is devoted to old and recent investigations of the classical M. A. Buhl problem of description of the compatible
linear vector field equations and their general M. G. Pfeiffer and modern Lax – Sato-type special solutions. In particular,
we analyze the related Lie-algebraic structures and the properties of integrability for a very interesting class of nonlinear
dynamical systems called the dispersion-free heavenly type equations, which were introduced by Pleba´nski and later
analyzed in a series of articles. The AKS-algebraic and related \scrR -structure schemes are used to study the orbits of the
corresponding coadjoint actions, which are intimately connected with the classical Lie – Poisson structures on them. It is
shown that their compatibility condition coincides with the corresponding heavenly type equations under consideration.
It is also demonstrated that all these equations are originated in this way and can be represented as a Lax compatibility
condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related
to the heavenly equations is described and its analytic structure connected with the Casimir invariants is indicated. In
addition, we present typical examples of equations of this kind demonstrating in detail their integrability via the scheme
proposed in the paper. The relationship between a very interesting Lagrange – d’Alembert-type mechanical interpretation
of the devised integrability scheme and the Lax – Sato equations is also discussed. |
| first_indexed | 2026-03-24T02:13:04Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-1811 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T02:13:04Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/49/efcdbe64e4fe560bc4f193225bdcf049 |
| spelling | umjimathkievua-article-18112019-12-05T09:28:05Z The classical M. A. Buhl problem, its Pfeiffer – Sato solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly type nonlinear equations Класична задача М. А. Буля, її розв’язки Пфайфера–Сато i класичний принцип Лагранжа–Даламбера для iнтегровних нелiнiйних рiвнянь небесного типу Prykarpatsky, Ya. A. Samoilenko, A. M. Прикарпатський, Я. А. Самойленко, А. М. The survey is devoted to old and recent investigations of the classical M. A. Buhl problem of description of the compatible linear vector field equations and their general M. G. Pfeiffer and modern Lax – Sato-type special solutions. In particular, we analyze the related Lie-algebraic structures and the properties of integrability for a very interesting class of nonlinear dynamical systems called the dispersion-free heavenly type equations, which were introduced by Pleba´nski and later analyzed in a series of articles. The AKS-algebraic and related \scrR -structure schemes are used to study the orbits of the corresponding coadjoint actions, which are intimately connected with the classical Lie – Poisson structures on them. It is shown that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is also demonstrated that all these equations are originated in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described and its analytic structure connected with the Casimir invariants is indicated. In addition, we present typical examples of equations of this kind demonstrating in detail their integrability via the scheme proposed in the paper. The relationship between a very interesting Lagrange – d’Alembert-type mechanical interpretation of the devised integrability scheme and the Lax – Sato equations is also discussed. Запропонований огляд присвячено попереднiм та новим дослiдженням класичної проблеми М. А. Буля опису узгоджених лiнiйних векторних полiв та її загальним розв’язкам М. Г. Пфайфера, а також новим спецiальним розв’язкам типу Лакса – Сато. Зокрема, проаналiзовано вiдповiднi алгебраїчнi структури Лi та властивостi iнтегровностi для дуже цiкавого класу нелiнiйних динамiчних систем, що називаються бездисперсiйними рiвняннями небесного типу, якi були введенi Є. Плебанським i пiзнiше проаналiзованi в серiї статей. На основi Лi-алгебраїчної схеми Адлера – Костанта – Сурiо (AKS) та асоцiйованої з нею $R$-структури вивчено орбiти вiдповiдних коприєднаних дiй, тiсно пов’язаних з клaсичними структурами Лi – Пуассона на них. Показано, що умова сумiсностi для них збiгається з вiдповiдними рiвняннями небесного типу, що аналiзуються. Також показано, що всi цi рiвняння породжуються таким чином i їх можна представити як умову сумiсностi Лакса для спецiально побудованих петельних векторних полiв на торi. Описано нескiнченну iєрархiю законiв збереження, пов’язаних з небесними рiвняннями, та вказано її аналiтичну структуру, що пов’язана з iнварiантами Казiмiра. Крiм того, наведено типовi приклади таких рiвнянь, для яких детально проаналiзовано їх iнтегровнiсть у рамках запропонованого пiдходу. Показано зв’язок розвинутого пiдходу з дуже цiкавою класичною механiчною iнтерпретацiєю Лагранжа – Даламбера. Institute of Mathematics, NAS of Ukraine 2017-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1811 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 12 (2017); 1652-1689 Український математичний журнал; Том 69 № 12 (2017); 1652-1689 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/1811/793 Copyright (c) 2017 Prykarpatsky Ya. A.; Samoilenko A. M. |
| spellingShingle | Prykarpatsky, Ya. A. Samoilenko, A. M. Прикарпатський, Я. А. Самойленко, А. М. The classical M. A. Buhl problem, its Pfeiffer – Sato solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly type nonlinear equations |
| title | The classical M. A. Buhl problem, its Pfeiffer – Sato
solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly
type nonlinear equations |
| title_alt | Класична задача М. А. Буля, її розв’язки Пфайфера–Сато
i класичний принцип Лагранжа–Даламбера для iнтегровних
нелiнiйних рiвнянь небесного типу |
| title_full | The classical M. A. Buhl problem, its Pfeiffer – Sato
solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly
type nonlinear equations |
| title_fullStr | The classical M. A. Buhl problem, its Pfeiffer – Sato
solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly
type nonlinear equations |
| title_full_unstemmed | The classical M. A. Buhl problem, its Pfeiffer – Sato
solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly
type nonlinear equations |
| title_short | The classical M. A. Buhl problem, its Pfeiffer – Sato
solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly
type nonlinear equations |
| title_sort | classical m. a. buhl problem, its pfeiffer – sato
solutions and the classical lagrange – d’alembert principle for the integrable heavenly
type nonlinear equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1811 |
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