On Symplectic and Hamiltonian Differential Operators

Abstract

The well-known papers [1,2] that revealed interrelations between integrability of the KdV equation and its Hamiltonian nature gave an impetus to the development of Hamiltonian theory of nonlinear evolution equations. Initially the researchers’ attention was focused on the concept of a Hamiltonian operator [3–6], that is the infinite-dimensional counterpart of the Poisson structure. Later it became clear that also important occur operators named symplectic [7,8], that are the inverses of the formally Hamiltonian nonlocal ones. At the present time the situation is that there are well-investigated classes of local Hamiltonian operators, such as the infinite-dimensional Kirillov — Kostant structures, quadratic Adler — Gelfand — Dikiĭ structure, Dubrovin — Novikov hydrodynamic structures, etc., and there are also some “solitary” examples of useful local symplectic operators. A unified approach developed by the author[7] presents both Hamiltonian and symplectic operators as particular cases of more general Dirac structures and shows that no preference for any of these two classes must be made.