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Dorfman, I. Ya.; Mokhov, O. I.

Local symplectic operators and structures related to them. (English) Zbl 0747.58028

J. Math. Phys. 32, No. 12, 3288-3296 (1991).
The interrelation between integrability and bi-Hamiltonian structures was studied for a long time. The authors initiate the study of the dual approach, integrability and bi-symplectic structures. In the case of infinite-dimensional space, the mentioned points of view prove to be quite different since local Poisson brackets are equivalent to nonlocal presymplectic structures and vice versa.
The article deals with symplecticity conditions for matrices whose entries are differential operators. It contains some classification theorems and many explicit examples. Also the interpretation by means of symplectic connections on loop spaces of pseudo-Riemannian manifolds, and the infinite-dimensional version of the Darboux theorem in terms of differential substitutions (Bäcklund transformations) are briefly mentioned.
Reviewer: J.Chrastina (Brno)

Cited in 3 Documents

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds

Keywords:

symplectic operator; Hamiltonian theory; Bäcklund transformations
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