TL;DR
This paper explores two similar extensions of the 1D Toda hierarchy, revealing their structural relationship and explaining the origin of the bilinear formalism's mysterious structure.
Contribution
It demonstrates that the extended Toda hierarchy and Ogawa's 2+1D extension are structurally similar, with the former as a dimensional reduction of the latter.
Findings
The two extensions have a very similar structure.
The extended Toda hierarchy can be viewed as a dimensional reduction.
This explains the origin of Milanov's bilinear formalism.
Abstract
The extended Toda hierarchy of Carlet, Dubrovin and Zhang is reconsidered in the 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.
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