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arXiv · 2409.13771

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

Jean-Pierre Magnot·Enrique G. Reyes
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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.

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Jean-Pierre Magnot, Enrique G. Reyes. 2024-09-18. Kadomtsev-Petviashvili hierarchies with non-formal pseudo-differential operators, non-formal solutions, and a Yang-Mills--like formulation. https://arxiv.org/abs/2409.13771

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