Old and New Reductions of Dispersionless Toda Hierarchy

[摘要] 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.

[关键词] dispersionless Toda hierarchy;finite-variable reduction;waterbag model;Landau–Ginzburg potential;L¨owner equations;Gibbons–Tsarev equations;hodograph solution;Darboux equations;Egorov metric;Combescure transformation;Frobenius manifold;flat coordinates