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

Abstract

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