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

For a scalar evolution equation $u_t=K(t,x,u,u_x,\dots ,u_n),n\ge 2$ the cohomology spaces $H^{1,s}({\mathcal{R}}^{\mathrm{\infty }})$ vanishes for $s\ge 3$ while the space $H^{1,2}({\mathcal{R}}^{\mathrm{\infty }})$ is isomorphic to the space of variational operators. The cohomology space $H^{1,2}({\mathcal{R}}^{\mathrm{\infty }})$ is also shown to be isomorphic to the space of symplectic operators for $u_t=K$ for which the equation is Hamiltonian. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The symplectic nature of the potential form of a bi-Hamiltonian evolution equation is also presented.