Reduction properties of the KP-mKP hierarchy

TL;DR

This paper investigates the reduction properties of the KP-mKP hierarchy, including its specific reductions and connections to other integrable hierarchies, confirming a conjecture related to the open KdV hierarchy.

Contribution

It extends understanding of the KP-mKP hierarchy's reduction properties and proves that Hirota equations of the extended r-reduced KP hierarchy derive from the mKP hierarchy, confirming a conjecture.

Findings

Reduction to KP, mKP, and BKP hierarchies.

Derivation of Hirota equations for extended r-reduced KP from mKP.

Confirmation of Alexandrov's conjecture on open KdV hierarchy.

Abstract

The so-called KP-mKP hierarchy, which was introduced recently via pseudo-differential operators with two derivations, can be reduced to the Kadomtsev-Petviashvili (KP), the modified KP (mKP) and the two-component BKP hierarchies. In this note, we continue to study reductions properties of the KP-mKP hierarchy, including its $(n,m)$-reduction and its reduction to a certain extended $r$-reduced KP hierarchy (the $r$-th Gelfand-Dickey together with its wave function). As a byproduct, we show that the Hirota equations of the extended $r$-reduced KP hierarchy follow from those of the mKP hierarchy, which confirms a conjecture of Alexandrov on the open KdV hierarchy in [ J. High Energy Phys. 2015 ].