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.