We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard–Magri scheme of integrability to a pair of compatible non-local Poisson structures. We apply this scheme to several such pairs, proving thereby integrability of various evolution equations, as well as hyperbolic equations.

中文翻译：

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非局部泊松结构及其在可积系统理论中的应用

我们建立在非局部泊松顶点代数概念上的严格的非局部泊松结构理论。作为一个应用程序，我们找到条件来保证Lenard-Magri方案的可整合性对一对兼容的非局部Poisson结构的适用性。我们将此方案应用于几个这样的对，从而证明了各种演化方程以及双曲方程的可积性。