Fels, M. E.Yaşar, E.2024-07-122024-07-122019-06-041402-9251https://doi.org/10.1080/14029251.2019.1640470https://www.atlantis-press.com/journals/jnmp/125950493https://hdl.handle.net/11452/43229For a scalar evolution equation u(t) = K(t, x, u, u(x), . . . , u(2m+1)) with m >= 1, the cohomology space H-1,H-2() is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for u(t) = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H-1,H-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.eninfo:eu-repo/semantics/closedAccessInverse problemCalculusBicomplexesVariational bicomplexCohomologyScalar evolution equationSymplectic operatorHamiltonian evolution equationMathematicsPhysicsArticle00047478010000760464926410.1080/14029251.2019.1640470