Journal
JOURNAL OF GEOMETRY AND PHYSICS
Volume 120, Issue -, Pages 106-128Publisher
ELSEVIER
DOI: 10.1016/j.geomphys.2017.05.010
Keywords
Classical integrability; Lax representation; R-matrix; Hierarchies
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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 Slclyanin characterized by the 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 sl(2, C) circle times C(lambda, lambda(-1)). The results are illustrated with the examples of the nonlinear Schrodinger and Gerdjikov-lvanov hierarchies. (C) 2017 Elsevier B.V. All rights reserved.
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