A general formula for the Magnus expansion in terms of iterated integrals of right-nested commutators

TL;DR

This paper derives a comprehensive formula for the Magnus expansion terms using iterated integrals of right-nested commutators, connecting it with permutation Hopf algebras.

Contribution

It introduces a general expression for Magnus series terms in terms of nested commutators and explores their relation to permutation Hopf algebras.

Findings

Provides a unified formula for Magnus expansion terms

Establishes a link between Magnus series and permutation Hopf algebras

Enhances understanding of algebraic structures in differential equations

Abstract

We present a general expression for any term of the Magnus series as an iterated integral of a linear combination of independent right-nested commutators with given coefficients. The relation with the Malvenuto--Reutenauer Hopf algebra of permutations is also discussed.