Old and New Reductions of Dispersionless Toda Hierarchy
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 L\"owner equations. Consistency of these L\"owner 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.
Keywords
Cite
@article{arxiv.1206.1151,
title = {Old and New Reductions of Dispersionless Toda Hierarchy},
author = {Kanehisa Takasaki},
journal= {arXiv preprint arXiv:1206.1151},
year = {2012}
}