Old and New Reductions of Dispersionless Toda Hierarchy

TL;DR

This paper explores geometric aspects of finite-variable reductions in the dispersionless Toda hierarchy, introducing new formulations using Landau-Ginzburg potentials and connecting them to classical differential geometry.

Contribution

It introduces two types of reductions using Landau-Ginzburg potentials and links them to L"owner and Gibbons-Tsarev equations, enriching the geometric understanding of the hierarchy.

Findings

Generalization of Dubrovin and Zhang's trigonometric polynomial reduction.

Introduction of a transcendental reduction resembling waterbag models.

Derivation of hodograph solutions via Gibbons-Tsarev equations.

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 L\"owner equations. Consistency of these L\"owner 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.