Abstract
We define and illustrate the novel notion of dual integrable hierarchies, on the example of the nonlinear Schrödinger (NLS) hierarchy. For each integrable nonlinear evolution equation (NLEE) in the hierarchy, dual integrable structures are characterized by the fact that the zerocurvature 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 biHamiltonian or generally multitime structure. The first formulation chooses purely spacedependent fields as configuration space; it yields the standard Poisson structure for NLS. The other one is new: it chooses purely timedependent fields as configuration space and yields a different Poisson structure at each level of the hierarchy. The corresponding NLEE becomes a 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 rmatrix structure, whereas traditionally only one of them is involved in the classical rmatrix 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 multidimensional lattice of Lax pair, through successive uses of the dual Poisson structures, is briefly introduced.
Original language  English 

Pages (fromto)  415–439 
Number of pages  25 
Journal  Nuclear Physics B 
Volume  902 
DOIs  
Publication status  Published  Jan 2016 
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Anastasia Doikou
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)