Berjawi S, Ferapontov E V, Kruglikov B, Novikov V
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK.
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa, Russia.
Proc Math Phys Eng Sci. 2020 Jan;476(2233):20190642. doi: 10.1098/rspa.2019.0642. Epub 2020 Jan 29.
We study second-order partial differential equations (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 non-trivial 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（PDEs），对于这些方程，由方程的特征簇定义的共形结构在每个解上都是半平坦的（自对偶或反自对偶）。我们证明了这一要求意味着蒙日 - 安培性质。由于共形结构的半平坦性等同于存在一个非平凡的无弥散拉克斯对，我们的结果解释了这样一个观察结果：在四维及更高维中，所有已知的标量二阶可积无弥散PDEs都是蒙日 - 安培型的。我们还得到了具有半平坦共形结构的四维蒙日 - 安培方程的一些部分分类结果。
