Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Fr\"olicher Lie groups

TL;DR

This paper establishes the well-posedness of the KP hierarchy using Fr"olicher and diffeological space frameworks, introduces a smooth Mulase factorization, and explores Hamiltonian structures and solutions with formal parameters.

Contribution

It constructs regular Fr"olicher Lie groups of pseudo-differential operators and provides two proofs of KP hierarchy well-posedness in a smooth setting, extending to formal parameter series.

Findings

Proves well-posedness of the KP hierarchy in a smooth category.

Introduces a smooth Mulase factorization for infinite-dimensional groups.

Derives Hamiltonian interpretation and solution sequences for KP equations.

Abstract

We recall the notions of Fr\"olicher and diffeological spaces and we build regular Fr\"olicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of the Mulase factorization of infinite dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also generalize these results to a KP hierarchy modelled on formal pseudo-differential operators with coefficients which are series in formal parameters, describe a rigorous derivation of the Hamiltonian interpretation of the KP hierarchy, and discuss how solutions depending on formal parameters can lead to sequences of functions converging to a class of solutions of the standard KP-I equation.