Kadomtsev-Petviashvili Hierarchies with Non-Formal Pseudo-Differential Operators, Non-Formal Solutions, and a Yang-Mills–Like Formulation

We start from the classical Kadomtsev-Petviashvili (KP) hierarchy posed on formal pseudo-differential operators, and we produce new hierarchies of non-linear equations in the context of non-formal pseudo-differential operators lying in the Kontsevich and Vishik s odd class. In particular, we show that it is possible to lift the standard KP hierarchy to hierarchies of differential equations for non-formal pseudo-differential operators, and to recover the former starting from the latter. 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 kind of solutions that they produce for the KP-II equation. © 2025 The Author(s)