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

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 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 ) ⊗ C ( λ , λ − 1 ). The results are illustrated with the examples of the nonlinear Schrödinger and Gerdjikov-Ivanov hierarchies.