Dispersionless version of multi-component Pfaff-Toda hierarchy

TL;DR

This work establishes that the N-component Pfaff-Toda hierarchy is equivalent to a 2N-component DKP hierarchy and derives its dispersionless limit in an elliptic uniformization. The dispersionless equations are encoded on an elliptic curve whose modular parameter $\tau$ becomes a dynamical variable driven by the times, and the uniformization expresses key quantities through Jacobi theta-functions. The authors present a compact elliptic formulation of the dispersionless hierarchy, derive the elliptic curve and its alternative representations, and translate the results to the Pfaff-Toda variables using the DKP–Pfaff-Toda equivalence. The findings connect dispersionless integrable systems with elliptic geometry and offer a framework for potential reductions and connections to BKP-type structures, with open questions about minimal generating relations and multi-component generalizations.

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.