Lagrangian and Hamiltonian structures in an integrable hierarchy and space-time duality

Jean Avan, Vincent Caudrelier, Anastasia Doikou, Anjan Kundu

Published 2015-10-05Version 1

We define and illustrate the novel notion of dual integrable hierarchies, on the example of the nonlinear Schr\"odinger (NLS) hierarchy. For each integrable nonlinear evolution equation (NLEE) in the hierarchy, dual integrable structures are characterized by the fact that the zero-curvature representation of the NLEE can be realized by two Hamiltonian formulations stemming from two inequivalent Poisson structures and two distinct Hamiltonians. This is different from the standard bi-Hamiltonian structure. One is well-known and based on the standard Poisson structure for NLS. The other one is new and based on a different Poisson structure at each level of the hierarchy, yielding the corresponding NLEE as a {\it space} evolution equation. We emphasize the role of the Lagrangian formulation as a unifying framework for deriving both Poisson structures, using ideas from covariant field theory. One of our main results is to show that the two matrices of the Lax pair satisfy the same form of ultralocal Poisson algebra (up to a sign) characterized by an $r$-matrix structure, whereas traditionally only one of them is involved in the classical $r$-matrix method. We construct explicit dual hierarchies of Hamiltonians, and Lax representations of the triggered dynamics, from the monodromy matrices of either Lax matrix. An appealing procedure to build a multi-dimensional lattice of Lax pair, through successive uses of the dual Poisson structures, is briefly introduced.