论文标题
在Pavlov-Mikhalev方程的对称性的递归操作员上
On recursion operators for symmetries of the Pavlov-Mikhalev equation
论文作者
论文摘要
在非线性偏微分方程的几何形状中，对方程式$ \ MATHCAL {E} $作用的递归运算符被理解为方程式$ \ MATHCAL {TE} $ cantent to $ \ Mathcal {e} $的Bäcklund自动转换。我们将这种方法应用于3D Pavlov-Mikhalev方程的天然两组分扩展 \ begin {equation*} u_ {yy} = u_ {tx} + u_yu_ {xx} - u_xu_ {xy}。 \ end {等式*}我们描述了该扩展的对称的谎言代数，构造了两个递归操作员（其中一个是一个较早的递归运算符）并找到其作用。我们还建立了这些操作员及其兼容性的世袭财产（从Frölicher-Nijenhuis bracket的意义上）。我们还发现了十二个其他操作员，这些操作员从某种意义上是退化的（我们称它们为\ emph {Queer}）并讨论其属性。在总结部分中，多维方程的两个组分保护定律的几何背景与其与差分覆盖率的关系相同。
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.
