标题： 与非形式伪微分算子、非形式解以及类似杨-米尔斯的表述相关的Kadomtsev-Petviashvili层次结构 显示英文标题

标题： Kadomtsev-Petviashvili hierarchies with non-formal pseudo-differential operators, non-formal solutions, and a Yang-Mills--like formulation

摘要： 我们从经典Kadomtsev-Petviashvili层次开始，该层次定义在形式伪微分算子上，并且我们生成了两个非线性方程层次，这些方程定义在位于Kontsevich和Vishik的奇类中的非形式伪微分算子上，其中一个层次的值在形式伪微分算子上。我们证明了相应的Zakharov-Shabat方程在此背景下成立，并将我们的一个层次表示为在伪微分联络空间上的一类Yang-Mills作用泛函的最小化，其曲率取值于Dixmier理想。最后，我们通过它们所产生的KP-II方程的解来比较我们的Kadomtsev-Petviashvili层次：存在性、唯一性和形式性。 显示英文摘要

摘要： We start from the classical Kadomtsev-Petviashvili hierarchy posed on formal pseudo-differential operators, and we produce two hierarchies of non-linear equations posed on non-formal pseudo-differential operators lying in the Kontsevich and Vishik's odd class, one of them with values in formal pseudo-differential operators. We prove that the corresponding Zakharov-Shabat equations hold in this context, and we express one of our hierarchies as the minimization of a class of Yang-Mills action functionals on a space of pseudo-differential connections whose curvature takes values in the Dixmier ideal. We finish by comparing our Kadomtsev-Petviashvili hierarchies in terms of the solutions that they produce to the KP-II equation: existence, uniqueness and formality.