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.

中文翻译：

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变分算子、辛算子和标量演化方程的上同调

对于 m ≥ 1 的标量演化方程 ut = K(t, x, u, ux, . . , u2m+1)，上同调空间 H1,2() 被证明与变分算子空间同构，并且给出了一个明确的同构。方程为哈密顿量的 ut = K 的辛算子空间也被证明与空间 H1,2() 同构，随后可以自然地与变分算子空间识别。三阶标量演化方程允许一阶辛（或变分）算子被表征。还介绍了双哈密尔顿演化方程的潜在形式的变分算子（或辛）性质，以生成感兴趣的示例。