1. Old and New Reductions of Dispersionless Toda Hierarchy
- Author
-
Kanehisa Takasaki
- Subjects
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 ,Mathematics ,QA1-939 - 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.
- Published
- 2012
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