Reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures

TL;DR

This paper investigates finite-dimensional reductions of the dispersionless 2D Toda hierarchy, demonstrating their connection to radial Loewner equations and constructing their Hamiltonian structures.

Contribution

It introduces a novel link between hierarchy reductions and radial Loewner equations and develops Hamiltonian structures using Ferapontov's approach.

Findings

Reductions are governed by radial Loewner equations.

Hamiltonian structures are explicitly constructed.

Provides a new framework for understanding hierarchy reductions.

Abstract

We study finite-dimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures, following an approach proposed by Ferapontov.