We recall the notions of Frölicher and diffeological spaces, and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of Mulase’s deep algebraic factorization of infinite-dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness of the Cauchy problem for the Kadomtsev–Petviashvili (KP) hierarchy in a smooth category. We also generalize these results to a KP hierarchy modelled on formal pseudo-differential operators with coefficients which are series in formal parameters, we describe a rigorous derivation of the Hamiltonian interpretation of the KP hierarchy, and we discuss how solutions depending on formal parameters can lead to sequences of functions converging to a class of solutions of the standard KP-II equation.

中文翻译：

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Kadomtsev–Petviashvili层次结构，Mulase因式分解和FrölicherLie组的良好定位

我们回想起Frölicher和差分空间的概念，并在一个自变量中建立正规的FrölicherLie群和形式化伪微分算子的Lie代数。将这些构造与基于形式的伪微分算子的无限维度组的Mulase深代数因式分解的平滑版本相结合，我们给出了Kadomtsev–Petviashvili（KP）层次结构在光滑的Cauchy问题的适定性的两个证明类别。我们还将这些结果推广到以形式为参数形式的系数的形式为形式的伪伪微分算子建模的KP层次结构，我们描述了对KP层次结构的哈密顿解释的严格推导，