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

TL;DR

This paper introduces the concept of dual integrable hierarchies, exemplified by the NLS hierarchy, revealing two distinct Hamiltonian formulations based on different configuration spaces, and explores their algebraic structures and implications.

Contribution

It defines and illustrates dual integrable hierarchies with two inequivalent Poisson structures, expanding the understanding of integrable systems beyond standard bi-Hamiltonian frameworks.

Findings

Both Lax matrices satisfy the same ultralocal Poisson algebra (up to a sign).

Explicit dual hierarchies of Hamiltonians and Lax representations are constructed.

A method for building multi-dimensional lattice of Lax pairs using dual Poisson structures is proposed.

Abstract

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 distinct choices of the configuration space, yielding two inequivalent Poisson structures on the corresponding phase space and two distinct Hamiltonians. This is fundamentally different from the standard bi-Hamiltonian or generally multitime structure. The first formulation chooses purely space-dependent fields as configuration space; it yields the standard Poisson structure for NLS. The other one is new: it chooses purely time-dependent fields as configuration space and yields a different Poisson structure at each level of the hierarchy. The corresponding NLEE becomes 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.