Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations. Issue 4 (2nd October 2019) - Fels, M.E.; Yaşar, E.
Abstract : For a scalar evolution equation ut = K ( t, x, u, ux, . . ., u 2 m +1 ) with m ≥ 1, the cohomology space H 1, 2 ( ) is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for ut = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H 1, 2 ( ) and subsequently can be naturally identified with the space of variational operators. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The variational operator (or symplectic) nature of the potential form of a bi-Hamiltonian evolution equation is also presented in order to generate examples of interest.