Old and New Reductions of Dispersionless Toda Hierarchy

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2012
• ISSN: 1815-0659
• 1. Verfasser: Takasaki, K.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2012
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/149183
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• 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.

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.