标题： 关于微分方程在射影变换群下不变的性质 显示英文标题

标题： On differential equations invariant under a projective transformation group

摘要： 我们考虑一个射影变换，并建立该变换群直到七阶的不变量。 我们利用所得的不变量构造了一类非线性发展方程，并在此类方程中识别出一些对称可积方程。 值得注意的是，在此类方程中，唯一的一个三阶对称可积演化方程是一个完全非线性方程，我们找到了该方程的递归算子及其与Schwarzian KdV的关系。 此外，我们还建立了此类方程中高阶对称可积方程属于完全非线性三阶方程族，并证明了五阶情况下的这一结论。 我们还识别出了在此射影变换下不变的常微分方程，并降低了这些方程的阶数。 显示英文摘要

摘要： We consider a projective transformation and establish the invariants for this transformation group up to order seven. We use the obtained invariants to construct a class of nonlinear evolution equations and identify some symmetry-integrable equations in this class. Notably, the only symmetry-integrable evolution equation of order three in this class is a fully-nonlinear equation for which we find the recursion operator and its connection to the Schwarzian KdV. We furthermore establish that higher-order symmetry-integrable equations in this class belong to the hierarchy of the fully-nonlinear 3rd-order equation and prove this for the 5th-order case. We also identify the ordinary differential equations that are invariant under this projective transformation and reduce the order of these equations.