Multicomponent DKP hierarchy and its dispersionless limit

TL;DR

This paper introduces the multicomponent DKP hierarchy using free fermions and bosonization, explores its dispersionless limit, and reveals an elliptic curve structure with a dynamical modulus.

Contribution

It presents the first formulation of the multicomponent DKP hierarchy via bilinear integral equations and analyzes its dispersionless limit with elliptic curve insights.

Findings

Hierarchy described by bilinear equations of Hirota-Miwa type

Elliptic curve structure with a dynamical modulus

Unified elliptic equation in the dispersionless limit

Abstract

Using the free fermions technique and bosonization rules we introduce the multicomponent DKP hierarchy as a generating bilinear integral equation for the tau-function. A number of bilinear equations of the Hirota-Miwa type are obtained as its corollaries. We also consider the dispersionless version of the hierarchy as a set of nonlinear differential equations for the dispersionless limit of logarithm of the tau-function (the $F$-function). We show that there is an elliptic curve built in the structure of the hierarchy, with the elliptic modulus being a dynamical variable. This curve can be uniformized by elliptic functions, and in the elliptic parametrization many dispersionless equations of the Hirota-Miwa type become equivalent to a single equation having a nice form.