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

TL;DR

This paper extends the Kadomtsev-Petviashvili hierarchy to non-formal pseudo-differential operators, introduces a Yang-Mills-like formulation, and analyzes solution properties such as existence and uniqueness.

Contribution

It develops new non-formal KP hierarchies within the Kontsevich and Vishik classes and formulates them using Yang-Mills action principles.

Findings

Established Zakharov-Shabat equations in the non-formal context

Expressed KP hierarchy as a Yang-Mills action minimization

Compared solution properties of different hierarchies for KP-II

Abstract

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.