Well-Posedness of the Kadomtsev–Petviashvili Hierarchy, Mulase Factorization, and Frölicher Lie Groups

Abstract

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of Mulase's deep algebraic 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, we describe a rigorous derivation of the Hamiltonian interpretation of the KP hierarchy, and we 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.

On the geometry of $Diff(S^1)$-pseudodifferential operators based on renormalized traces.

“…One then can consider "natural" notions of smoothness, inherited from the embedding into Cl(S 1 , V ) for the pseudo-differential part, and from the well-known structure of ILB Lie group [31] from the diffeomorphism (phase) component. In order to be more rigorous, one can then consider Frölicher Lie groups along the lines of [23,25] in this context, or in [24,27] when dealing with other examples where this setting is useful. A not-so-complete description of technical properties of Frölicher Lie groups can be found in works by other authors [16,29,30] but this area of knowledge, however, still needs to be further developed.…”

“…Due to the presence of unbounded pseudo-differential operators, and in particular differential operators of order 1, the Lie algebra of this group cannot be embedded in a group of bounded operators acting on the space of sections C 8 (S 1 , V ), but only represented in it. As a technical remark, we have to say that we have here an example of non-regular infinite dimensional Lie group, as given in Remark 2.31 adapting a remark from [27]. But another technical feature is that this group does not seem to carry atlas, for the same reasons of presence of unbounded operators, as first described in [2] in the context of formal pseudo-differential operators.…”

“…But one can anyway consider this group as an "infinite dimensional group" in the way of [12], or as a so-called Frölicher Lie group, see e.g. [27] and references therein, in order to make safe the notion of smoothness. In the section dedicated to the preliminaries, we describe this groups and some of its remarkable subgroups, we recall classical properties of the (zeta-)renormalized traces of pseudo-differential operators, describe useful splittings of the algebra of formal pseudo-differential operators from existing literature, and fully describe the space of formal Diff(S 1 )-pseudo-differential operators.…”

Rigid body equations on spaces of pseudo-differential operators with renormalized trace

“…Regularity is reviewed in [21,22] and also in Paycha's lectures, see [27, p. 95]. The Lie group structure of Cl 0, * (M, E) is discussed in [27,Proposition 4].…”

On the differential geometry of numerical schemes and weak solutions of functional equations

“…group equipped with a diffeology which makes composition and inversion smooth), see e.g. [33], as smooth vector space.…”