On the origin of dual Lax pairs and their $r$-matrix structure

TL;DR

This paper explores the algebraic foundations of dual Lax pairs in integrable systems, revealing their common $r$-matrix structure through Lie-Poisson brackets and illustrating with nonlinear Schrödinger and Gerdjikov-Ivanov hierarchies.

Contribution

It introduces the concept of dual Lax pairs and Hamiltonian formulations, linking their $r$-matrix structures to a single Lie-Poisson bracket on a coadjoint orbit.

Findings

Dual Hamiltonian formulations exist for integrable PDEs.

A shared $r$-matrix structure underpins dual Lax pairs.

Examples include nonlinear Schrödinger and Gerdjikov-Ivanov hierarchies.

Abstract

We establish the algebraic origin of the following observations made previously by the authors and coworkers: (i) A given integrable PDE in $1+1$ dimensions within the Zakharov-Shabat scheme related to a Lax pair can be cast in two distinct, dual Hamiltonian formulations; (ii) Associated to each formulation is a Poisson bracket and a phase space (which are not compatible in the sense of Magri); (iii) Each matrix in the Lax pair satisfies a linear Poisson algebra a la Sklyanin characterized by the {\it same} classical $r$ matrix. We develop the general concept of dual Lax pairs and dual Hamiltonian formulation of an integrable field theory. We elucidate the origin of the common $r$-matrix structure by tracing it back to a single Lie-Poisson bracket on a suitable coadjoint orbit of the loop algebra ${\rm sl}(2,\CC) \otimes \CC (\lambda, \lambda^{-1})$. The results are illustrated with the examples of the nonlinear Schr\"odinger and Gerdjikov-Ivanov hierarchies.