Construction of KP Hierarchies in Terms of Finite Number of Fields and their Abelianization

TL;DR

This paper develops multi-boson representations of the KP hierarchy using finite fields, providing a Poisson reduction framework and a finite-field formulation with Darboux coordinates, enriching the algebraic structure understanding.

Contribution

It introduces a construction of KP hierarchy representations with finite fields and describes their algebraic structure via Poisson reductions and Darboux coordinates.

Findings

Constructed $2M$-boson representations from $M$ two-boson KP representations.

Established multi-boson representations as Poisson reductions within the $R$-matrix scheme.

Provided a finite-field formulation of KP hierarchy in Darboux coordinates.

Abstract

The $2M$-boson representations of KP hierarchy are constructed in terms of $M$ mutually independent two-boson KP representations for arbitrary number $M$. Our construction establishes the multi-boson representations of KP hierarchy as consistent Poisson reductions of standard KP hierarchy within the $R$-matrix scheme. As a byproduct we obtain a complete description of any finitely-many-field formulation of KP hierarchy in terms of Darboux coordinates with respect to the first Hamiltonian structure. This results in a series of representations of $\Win1\,$ algebra made out of arbitrary even number of boson fields.