Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5499926 | Journal of Geometry and Physics | 2017 | 23 Pages |
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 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)âC(λ,λâ1). The results are illustrated with the examples of the nonlinear Schrödinger and Gerdjikov-Ivanov hierarchies.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematical Physics
Authors
Jean Avan, Vincent Caudrelier,