Two extensions of 1D Toda hierarchy

Abstract

The extended Toda hierarchy of Carlet, Dubrovin and Zhang is reconsidered in light of a 2 + 1D extension of the 1D Toda hierarchy constructed by Ogawa. These two extensions of the 1D Toda hierarchy turn out to have a very similar structure, and the former may be thought of as a kind of dimensional reduction of the latter. In particular, this explains an origin of the mysterious structure of the bilinear formalism proposed by Milanov.