Calculations needed for the proof of Theorem 1 (Mathematica file Computation Th1.nb and its description Computation Th1.pdf) from Second-order PDEs in four dimensions with half-flat conformal structure

We study second-order PDEs in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a nontrivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.

我们研究四维空间中的二阶偏微分方程（second-order PDEs），这类方程在每个解上由方程特征簇定义的共形结构均为半平坦（自对偶或反自对偶）形式。我们证明了这一条件蕴含蒙日-安培（Monge–Ampère）性质。由于共形结构的半平坦性等价于非平凡无色散Lax对（dispersionless Lax pair）的存在性，我们的结果解释了如下观测现象：四维及更高维数下所有已知的标量二阶可积无色散偏微分方程均属于蒙日-安培型。此外，我们还得到了若干四维空间中具有半平坦共形结构的蒙日-安培方程的部分分类结果。