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

2024/09/18 by Jean-Pierre Magnot, Magnot, Jean-Pierre, Enríque G. Reyes +1 · 1 citation
Mathematics · Physics and Astronomy · #35Q51 #37K10 #37K25 #37K30 #47N20 #58B25 #58J40 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2409.13771

openalex publication_date 2024/09/18 · openalex created_date 2024/10/25 · openalex updated_date 2026/08/01

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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