The Recursion operators of the BKP hierarchy and the CKP Hierarchy

TL;DR

This paper derives explicit recursion operators for the BKP and CKP hierarchies under (2n+1)-reduction, highlighting differences in their flow equations and demonstrating their application through explicit examples.

Contribution

It introduces a new operator B to express odd variables via even variables and constructs formal recursion operators for BKP and CKP hierarchies under (2n+1)-reduction.

Findings

Recursion operators explicitly constructed for 3-reduction cases.

Flow equations differ between BKP and CKP hierarchies.

Generated flows are consistent with flow equations.

Abstract

In this paper, under the constraints of the BKP(CKP) hierarchy, a crucial observation is that the odd dynamical variable $u_{2k+1}$ can be explicitly expressed by the even dynamical variable $u_{2k}$ in the Lax operator $L$ through a new operator $B$. Using operator $B$, the essential differences between the BKP hierarchy and the CKP hierarchy are given by the flow equations and the recursion operators under the $(2n+1)$-reduction. The formal formulas of the recursion operators for the BKP and CKP hierarchy under $(2n+1)$-reduction are given. To illustrate this method, the two recursion operators are constructed explicitly for the 3-reduction of the BKP and CKP hierarchies. The $t_7$ flows of $u_2$ are generated from $t_1$ flows by the above recursion operators, which are consistent with the corresponding flows generated by the flow equations under 3-reduction.