The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups

TL;DR

This paper explores the Lax equation within infinite-dimensional Lie algebras, deriving explicit solutions and formulas, and establishing regularity properties for associated Lie groups, especially in the asymptotic estimate setting.

Contribution

It extends the Baker-Campbell-Dynkin-Hausdorff formula to product integrals in infinite-dimensional Lie groups and introduces an integral transformation to analyze regularity.

Findings

Derived an explicit integral expansion for solutions of the Lax equation.

Generalized the BCH formula to product integrals in infinite-dimensional Lie groups.

Proved regularity results for asymptotic estimate Lie groups using a novel integral transformation.

Abstract

We investigate the Lax equation in the context of infinite-dimensional Lie algebras. Explicit solutions are discussed in the sequentially complete asymptotic estimate context, and an integral expansion (sums of iterated Riemann integrals over nested commutators with correction term) is derived for the situation that the Lie algebra is inherited by an infinite-dimensional Lie group in Milnor's sense. In the context of Banach Lie groups (and Lie groups with suitable regularity properties), we generalize the Baker-Campbell-Dynkin-Hausdorff formula to the product integral (with additional nilpotency assumption in the non-Banach case). We combine this formula with the results obtained for the Lax equation to derive an explicit representation of the product integral in terms of the exponential map. An important ingredient in the non-Banach case is an integral transformation that we introduce. This transformation maps continuous Lie algebra-valued curves to smooth ones and leaves the product integral invariant. This transformation is also used to prove a regularity statement in the asymptotic estimate context.