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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| Format: | Artikel |
| Sprache: | English |
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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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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 |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Old and New Reductions of Dispersionless Toda Hierarchy |
| spellingShingle |
Old and New Reductions of Dispersionless Toda Hierarchy Takasaki, K. |
| title_short |
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_sort |
old and new reductions of dispersionless toda hierarchy |
| author |
Takasaki, K. |
| author_facet |
Takasaki, K. |
| publishDate |
2012 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149183 |
| 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. |
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