Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 102, 22 pages      arXiv:1206.1151      https://doi.org/10.3842/SIGMA.2012.102
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Old and New Reductions of Dispersionless Toda Hierarchy

Kanehisa Takasaki
Graduate School of Human and Environmental Studies, Kyoto University, Yoshida, Sakyo, Kyoto, 606-8501, Japan

Received June 06, 2012, in final form December 15, 2012; Published online December 19, 2012

Abstract
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.

Key words: dispersionless Toda hierarchy; finite-variable reduction; waterbag model; Landau-Ginzburg potential; Löwner equations; Gibbons-Tsarev equations; hodograph solution; Darboux equations; Egorov metric; Combescure transformation; Frobenius manifold; flat coordinates.

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