一类允许伙伴对称的二阶偏微分方程
On a class of second-order PDEs admitting partner symmetries
摘要：
最近，我们已经演示了如何使用伙伴对称性来获得普列班斯基天体方程的非不变解，这些方程支配着天体引力度规。在本文中，我们提出了一类具有四个变量的标量二阶偏微分方程，具有伙伴对称性，只包含未知数的二阶导数。我们提出了一个一般形式的这样的偏微分方程连同伙伴对称之间的递归关系。将一般偏微分方程转化为几个最简单的标准型，其中包括两个Plebanski天体方程和另外两个非线性方程，我们称之为混合天体方程和非对称天体方程。在一个例子的混合天球方程，我们展示了如何使用伙伴对称性获得非不变解的偏微分方程的升降机不变的解决方案。最后，我们提出的Ricci平坦的自对偶度规的解决方案的混合天球方程和它的勒让德变换。
英文摘要：
Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs with four variables, that possess partner symmetries and contain only second derivatives of the unknown. We present a general form of such a PDE together with recursion relations between partner symmetries. This general PDE is transformed to several simplest canonical forms containing the two heavenly equations of Plebanski among them and two other nonlinear equations which we call mixed heavenly equation and asymmetric heavenly equation. On an example of the mixed heavenly equation, we show how to use partner symmetries for obtaining noninvariant solutions of PDEs by a lift from invariant solutions. Finally, we present Ricci-flat self-dual metrics governed by solutions of the mixed heavenly equation and its Legendre transform.
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。
