1206.1151 (Kanehisa Takasaki)

Old and new reductions of dispersionless Toda hierarchy    [PDF]

Kanehisa Takasaki

Two types of finite-variable reductions of the dispersionless Toda hierarchy are considered in the geometric perspectives. 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 intended to be a Toda version of the waterbag model of the dispersionless KP hierarchy. The two types of Landau-Ginzburg potentials are shown to satisfy (a radial version of) the L\"onwer equations with respect to the critical values of the Landau-Ginzburg potentials. Integrability conditions of these L\"owner equations are (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). Frobenius structures on the parameter space of the Landau-Ginzburg potentials are introduced, and flat coordinates are constructed explicitly.

View original: http://arxiv.org/abs/1206.1151