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 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.

Keywords

• 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