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arXiv 1902.08178math.DGmath-phmath.MPnlin.SI
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变分算子、辛算子与标量发展方程的上同调

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

Mark E. Fels, Emrullah Yasar

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42 pages
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摘要：

对于一个标量发展方程$u_t=K(t,x,u,u_x,\ldots, u_n), n\geq 2$，上同调空间$H^{1,s}({\mathcal R}^\infty)$对于$s\geq 3$为零，而空间$H^{1,2}({\mathcal R}^\infty)$同构于变分算子空间.上同调空间$H^{1,2}({\mathcal R}^\infty)$也被证明是同构的空间的辛算子$u_t=K$的方程是哈密顿。刻画了含有一阶辛（或变分）算子的三阶标量发展方程。本文还讨论了双Hamilton发展方程的势形式的辛性质。

英文摘要：

For a scalar evolution equation $u_t=K(t,x,u,u_x,\ldots, u_n), n\geq 2$ the cohomology spaces $H^{1,s}({\mathcal R}^\infty)$ vanishes for $s\geq 3$ while the space $H^{1,2}({\mathcal R}^\infty)$ is isomorphic to the space of variational operators. The cohomology space $H^{1,2}({\mathcal R}^\infty)$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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