关于Pavlov-Mikhalev方程对称性的递归算子
On recursion operators for symmetries of the Pavlov-Mikhalev equation
摘要：
在非线性偏微分方程几何中，作用于方程$\mathcal{E}$对称性的递归算子被理解为方程$\mathcal{TE}$与$\mathcal{E}$相切的Bäcklund自变换。我们将这种方法应用于自然的两个组件的3D巴甫洛夫-Mikhalev方程\begin{equation*} u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}. \end{equation*}的扩展。我们描述了李代数的对称性，这种扩展，构造两个递归算子（其中一个是已知的），并找到他们的行动。我们还建立了这些运营商的遗传性以及它们的兼容性（在Frölicher-Nijenhuis括号的意义上）。我们还发现12个额外的运营商是退化的意义上（我们称之为\emph{queer}），并讨论其性质。在结论部分中，揭示了多维方程二分量守恒律的几何背景及其与微分覆盖的关系。
英文摘要：
In geometry of nonlinear partial differential equations, recursion operators that act on symmetries of an equation $\mathcal{E}$ are understood as Bäcklund auto-transformations of the equation $\mathcal{TE}$ tangent to $\mathcal{E}$. We apply this approach to a natural two-component extension of the 3D Pavlov-Mikhalev equation \begin{equation*} u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}. \end{equation*} We describe the Lie algebra of symmetries for this extension, construct two recursion operators (one of them was known earlier) and find their action. We also establish the hereditary property of these operators as well as their compatibility (in the sense of the Frölicher-Nijenhuis bracket). We find also twelve additional operators which are degenerate in a sense (we call them \emph{queer}) and discuss their properties. In the concluding part, a geometrical background of two-component conservation laws for multi-dimensional equations is exposed together with its relations to differential coverings.
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