Additional symmetries of constrained CKP and BKP hierarchies

TL;DR

This paper investigates the additional symmetries of constrained CKP and BKP hierarchies, revealing their algebraic structures and explicit actions on eigenfunctions, and introduces operators to define these symmetries.

Contribution

It provides explicit descriptions of additional symmetries for constrained CKP and BKP hierarchies and introduces operators to define these symmetry flows.

Findings

Symmetries act on Lax operators and eigenfunctions.

New subalgebras of $W_{1+ ablafty}$ are formed.

Operators $Y_k$ are introduced to define symmetry flows.

Abstract

The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP) hierarchies are given by their actions on the Lax operators, and their actions on the eigenfunction and adjoint eigenfunction $\{\Phi_i,\Psi_i \}$ are presented explicitly. Furthermore, we show that acting on the space of the wave operator, $\partial_k^*$ forms new centerless $W^{cC}_{1+\infty}$ and $W^{cB}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$ respectively. In order to define above symmetry flows $\partial_k^*$ of the cCKP and cBKP hierarchies, two vital operators $Y_k$ are introduced to revise the additional symmetry flows of the CKP and BKP hierarchies.