Dispersionless version of multi-component Pfaff-Toda hierarchy

TL;DR

This paper studies the dispersionless limit of the multi-component Pfaff-Toda hierarchy, revealing its connection to elliptic curves and simplifying its analysis through a reduction to the DKP hierarchy.

Contribution

It introduces the dispersionless version of the multi-component Pfaff-Toda hierarchy and shows its equivalence to a reduced DKP hierarchy, with a novel elliptic curve structure.

Findings

Hierarchy reduces to nonlinear differential equations for the tau-function's logarithm.

Elliptic curve structure with a dynamical modular parameter is embedded in the hierarchy.

Elliptic parametrization simplifies the hierarchy's form.

Abstract

We consider the dispersionless limit of the recently introduced multi-component Pfaff-Toda hierarchy. Its dispersionless version is a set of nonlinear differential equations for the dispersionless limit of logarithm of the tau-function (the F-function). They are obtained as limiting cases of bilinear equations of the Hirota-Miwa type. The analysis of the Pfaff-Toda hierarchy is substantially simplified by using the observation that the full (not only dispersionless) N-component Pfaff-Toda hierarchy is actually equivalent to the 2N-component DKP hierarchy. In the dispersionless limit, there is an elliptic curve built in the structure of the hierarchy, with the elliptic modular parameter being a dynamical variable. This curve can be uniformized by elliptic functions, and in the elliptic parametrization the hierarchy acquires a compact and especially nice form.