%0 Journal Article
%T Old and New Reductions of Dispersionless Toda Hierarchy
%A Kanehisa Takasaki
%J Symmetry, Integrability and Geometry : Methods and Applications
%D 2012
%I National Academy of Science of Ukraine
%X 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 wner equations. Consistency of these L wner 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.
%K dispersionless Toda hierarchy
%K finite-variable reduction
%K waterbag model
%K Landau-Ginzburg potential
%K L wner equations
%K Gibbons-Tsarev equations
%K hodograph solution
%K Darboux equations
%K Egorov metric
%K Combescure transformation
%K Frobenius manifold
%K flat coordinates
%U http://dx.doi.org/10.3842/SIGMA.2012.102