Old and New Reductions of Dispersionless Toda Hierarchy
This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of ''Landau-Ginzburg potentials'' that play the role of reduced Lax functions. One of them is a generaliza...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2012 |
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Інститут математики НАН України
2012
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/149183 |
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| Zitieren: | Old and New Reductions of Dispersionless Toda Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 37 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862700674609840128 |
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| author | Takasaki, K. |
| author_facet | Takasaki, K. |
| citation_txt | Old and New Reductions of Dispersionless Toda Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 37 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of ''Landau-Ginzburg potentials'' that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric polynomial. The other is a transcendental function, the logarithm of which resembles the waterbag models of the dispersionless KP hierarchy. They both satisfy a radial version of the Löwner equations. Consistency of these Löwner equations yields a radial version of the Gibbons-Tsarev equations. These equations are used to formulate hodograph solutions of the reduced hierarchy. Geometric aspects of the Gibbons-Tsarev equations are explained in the language of classical differential geometry (Darboux equations, Egorov metrics and Combescure transformations). Flat coordinates of the underlying Egorov metrics are presented.
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| first_indexed | 2025-12-07T16:39:41Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149183 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:39:41Z |
| publishDate | 2012 |
| publisher | Інститут математики НАН України |
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| spelling | Takasaki, K. 2019-02-19T18:21:19Z 2019-02-19T18:21:19Z 2012 Old and New Reductions of Dispersionless Toda Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q99; 37K10; 53B50; 53D45 DOI: http://dx.doi.org/10.3842/SIGMA.2012.102 https://nasplib.isofts.kiev.ua/handle/123456789/149183 This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of ''Landau-Ginzburg potentials'' that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric polynomial. The other is a transcendental function, the logarithm of which resembles the waterbag models of the dispersionless KP hierarchy. They both satisfy a radial version of the Löwner equations. Consistency of these Löwner equations yields a radial version of the Gibbons-Tsarev equations. These equations are used to formulate hodograph solutions of the reduced hierarchy. Geometric aspects of the Gibbons-Tsarev equations are explained in the language of classical differential geometry (Darboux equations, Egorov metrics and Combescure transformations). Flat coordinates of the underlying Egorov metrics are presented. This paper is a contribution to the Special Issue “Geometrical Methods in Mathematical Physics”. The full collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html.
 We thank the referees for many valuable comments. This work is partly supported by JSPS Grants-in-Aid for Scientific Research No. 21540218 and No. 22540186 from the Japan Society for the Promotion of Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Old and New Reductions of Dispersionless Toda Hierarchy Article published earlier |
| spellingShingle | Old and New Reductions of Dispersionless Toda Hierarchy Takasaki, K. |
| title | Old and New Reductions of Dispersionless Toda Hierarchy |
| title_full | Old and New Reductions of Dispersionless Toda Hierarchy |
| title_fullStr | Old and New Reductions of Dispersionless Toda Hierarchy |
| title_full_unstemmed | Old and New Reductions of Dispersionless Toda Hierarchy |
| title_short | Old and New Reductions of Dispersionless Toda Hierarchy |
| title_sort | old and new reductions of dispersionless toda hierarchy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149183 |
| work_keys_str_mv | AT takasakik oldandnewreductionsofdispersionlesstodahierarchy |