关于最近引入的一个五阶双Hamilton方程及其平凡相关的Hamilton算子
On a Recently Introduced Fifth-Order Bi-Hamiltonian Equation and Trivially Related Hamiltonian Operators
摘要：
我们证明了最近引进的一个五阶双哈密顿方程的微分约束任意函数的A。de Sole，V.G. Kac和M. Wakimoto不是一个新的方程，而是一个三阶方程的更高对称性。我们给出了这个方程中任意函数的一个详尽的例子，在每个例子中，相关的方程都不等价于其余例子中的方程。在每一种情况下的方程都与文献中已知的方程通过可逆变换联系起来。它表明，新的七阶哈密顿算子，使用它得到的引入的方程，是平凡相关的一对已知的五阶和三阶相容的哈密顿算子。利用低阶Hamilton算子的平凡复合，给出了高阶Hamilton算子的非局部推广。
英文摘要：
We show that a recently introduced fifth-order bi-Hamiltonian equation with a differentially constrained arbitrary function by A. de Sole, V.G. Kac and M. Wakimoto is not a new one but a higher symmetry of a third-order equation. We give an exhaustive list of cases of the arbitrary function in this equation, in each of which the associated equation is inequivalent to the equations in the remaining cases. The equations in each of the cases are linked to equations known in the literature by invertible transformations. It is shown that the new Hamiltonian operator of order seven, using which the introduced equation is obtained, is trivially related to a known pair of fifth-order and third-order compatible Hamiltonian operators. Using the so-called trivial compositions of lower-order Hamiltonian operators, we give nonlocal generalizations of some higher-order Hamiltonian operators.
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