On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

We establish a rigorous link between infinite-dimensional regular Frolicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a (parameter-depending) version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator of our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral operators in which this correspondence seems to take place. Also, motivated by Mulase’s work on the KP hierarchy, we prove a group factorization theorem for this group of Fourier integral operators.

Keywords

Kadomtsev-Petviashvili hierarchy, Mulase factorization, infinite jets, Frechet Lie groups, Frolicher Lie groups, Fourier integral operators, odd-class pseudodifferential operators

2010 Mathematics Subject Classification

Primary 35Q51, 37K10, 37K25, 37K30, 58J40. Secondary 47N20, 58B25.

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Received 26 April 2021

Published 21 May 2024