Gauge transformation and symmetries of the commutative multi-component BKP hierarchy

TL;DR

This paper introduces a new multi-component BKP hierarchy within a commutative algebra, explores its gauge transformations, constructs a constrained hierarchy with integrable systems, and establishes quantum torus symmetries and constraints.

Contribution

It defines a novel commutative multi-component BKP hierarchy, develops its gauge transformations, constructs a constrained hierarchy with integrable systems, and uncovers quantum torus symmetries.

Findings

Construction of a new commutative multi-component BKP hierarchy

Development of gauge transformation framework for the hierarchy

Identification of quantum torus symmetry and constraints

Abstract

In this paper, we defined a new multi-component BKP hierarchy which takes values in a commutative subalgebra of $gl(N,\mathbb C)$. After this, we give the gauge transformation of this commutative multi-component BKP (CMBKP) hierarchy. Meanwhile we construct a new constrained CMBKP hierarchy which contains some new integrable systems including coupled KdV equations under a certain reduction. After this, the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi-component BKP hierarchy will be constructed.