R-Matrix Formulation of KP Hierarchies and their Gauge Equivalence

TL;DR

This paper uses the R-matrix approach to show the equivalence of KP and its modifications, establishing a gauge transformation that links their algebraic structures and introduces a new bosonic field representation.

Contribution

It presents a novel R-matrix formulation demonstrating the gauge equivalence of KP hierarchies and introduces a new bosonic field representation of W-infinity algebras.

Findings

All three hierarchies are shown to be equivalent.

A symplectic gauge transformation connects the hierarchies.

New representation of W-infinity algebras with 4 bosonic fields.

Abstract

The Adler-Kostant-Symes $R$-bracket scheme is applied to the algebra of pseudo-differential operators to relate the three integrable hierarchies: KP and its two modifications, known as nonstandard integrable models. All three hierarchies are shown to be equivalent and connection is established in the form of a symplectic gauge transformation. This construction results in a new representation of the W-infinity algebras in terms of 4 bosonic fields.