Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions

TL;DR

This paper demonstrates the equivalence of four formulations of the Modified KP hierarchy, explores its reductions, and provides a simple explicit description of all polynomial tau-functions across related hierarchies.

Contribution

It introduces four equivalent formulations of the MKP hierarchy and offers a novel, simple description of all polynomial tau-functions for multiple integrable hierarchies.

Findings

Four equivalent formulations of the MKP hierarchy.

Explicit description of polynomial tau-functions for KP, MKP, and n-KdV hierarchies.

Insights into reductions to modified n-KdV hierarchies.

Abstract

We give 4 formulations of the Modified KP hierarchy and show that they are equivalent. We also discuss the reductions of the MKP hierarchy to the modified $n$-KdV hierarchies. As a byproduct, we find an astonishingly simple explicit description of all polynomial tau-functions of the KP, the MKP and the $n$-KdV hierarchies, and (implicitly) also for the modified $n$-KdV hierarchy.