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

摘要

AbstractWe establish the algebraic origin of the following observations made previously by the authors and coworkers: (i) A given integrable PDE in1+1dimensions 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 thesameclassicalrmatrix. We develop the general concept of dual Lax pairs and dual Hamiltonian formulation of an integrable field theory. We elucidate the origin of the commonr-matrix structure by tracing it back to a single Lie–Poisson bracket on a suitable coadjoint orbit of the loop algebrasl(2,C)?C(λ,λ?1). The results are illustrated with the examples of the nonlinear Schr?dinger and Gerdjikov–Ivanov hierarchies.]]>