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For a scalar evolution equation ut = K(t, x, u, ux, . . . , u2m+1) with m ≥ 1, the cohomology space H1,2() 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 H1,2() 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.

Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations