The applications of the gauge transformation for the BKP hierarchy

TL;DR

This paper explores four applications of gauge transformations in the BKP hierarchy, revealing connections to Toda lattice structures, Pfaffian tau functions, Fay-like identities, and symmetry compatibility.

Contribution

It introduces novel applications of gauge transformations in the BKP hierarchy, including structural, functional, and symmetry-related insights.

Findings

Gauge transformation orbit defines a (2+1)-dimensional Toda lattice structure.

Tau function generated by gauge transformation is a Pfaffian.

Higher Fay-like identities for BKP are derived using gauge transformations.

Abstract

In this paper, we investigated four applications of the gauge transformation for the BKP hierarchy. Firstly, it is found that the orbit of the gauge transformation for the constrained BKP hierarchy defines a special $(2 +1)$-dimensional Toda lattice equation structure. Then the tau function of the BKP hierarchy generated by the gauge transformation is showed to be the Pfaffian. And the higher Fay-like identities for the BKP hierarchy is also obtained through the gauge transformation. At last, the compatibility between the additional symmetry and the gauge transformation of the BKP hierarchy is proved.