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

Abstract

We establish a rigorous link between infinite-dimensional regular Frölicher 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. Contents 1. Introduction 236 2. Preliminaries on categories of regular Frölicher Lie groups 238 3. Preliminaries on pseudodifferential operators 243 4. The h-KP hierarchy with non-formal odd-class operators 246 5. KP equations and Dif f + (S 1 ) 253 Appendix:the group of Dif f + (S 1 )−pseudodifferential operators 254 References 258