The "Ghost" Symmetry of the BKP hierarchy

TL;DR

This paper develops the 'ghost' symmetry for the BKP hierarchy, revealing new spectral representations, potentials, and re-deriving key identities, advancing understanding of integrable systems.

Contribution

It introduces the spectral representation of eigenfunctions and a new potential using squared eigenfunction potential, and re-derives fundamental identities via 'ghost' symmetry.

Findings

Spectral representation of eigenfunctions established

New potential introduced via squared eigenfunction potential

Re-derivation of bilinear identity and Adler-Shiota-van-Moerbeke formula

Abstract

In this paper, we systematically develop the "ghost" symmetry of the BKP hierarchy through its actions on the Lax operator $L$, the eigenfunctions and the $\tau$ function. In this process, the spectral representation of the eigenfunctions and a new potential are introduced by using squared eigenfunction potential(SEP) of the BKP hierarchy. Moreover, the bilinear identity of the constrained BKP hierarchy and Adler-Shiota-van-Moerbeke formula of the BKP hierarchy are re-derived compactly by means of the spectral representation and "ghost" symmetry.