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

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 ${\rm sl}(2,\CC) \otimes \CC (λ, λ^{-1})$. The results are illustrated with the examples of the nonlinear Schrödinger and Gerdjikov-Ivanov hierarchies.

36 pages, 1 figure. Final version in line with published article. The latter is available for free at https://authors.elsevier.com/a/1VILr1PNYn24TG (until August 18 2017)

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