Alternative representation of Magnus series by exact proper operator exponent

TL;DR

This paper presents an exact operator exponential approach to representing the Magnus series, enabling straightforward analytical solutions for linear and nonlinear differential equations through series resummation.

Contribution

It introduces a novel method converting Dyson's time-ordered solutions into simple operator exponentials using BCH and Zassenhaus formulas, facilitating easier solution calculations.

Findings

Exact operator exponential representation of Magnus series.

Simplified analytical solution expressions as Taylor series.

New approach for resumming series in differential equations.

Abstract

In this report the emphasis is on an alternative representation of the Magnus series by proper operator (matrix) exponential solutions to differential equations (systems), both linear and nonlinear ODEs and PDEs. The main idea here is in \emph{exact} \emph{linear} representations of the \emph{nonlinear} DEs. We proceeded from Dyson's time-ordered solutions, and using only generalizations of the well-known Baker-Campbell- Hausdorff (BCH) and Zassenhaus formulae for $t$-dependent operators directly converted them to simple proper operator exponents. The method being explicit both in terms of the operator and in terms of expressing the formal solution as an ordinary exponential, makes it quite easy to calculate analytical expressions to solutions in the form of a Taylor function series in one variable $t$. If introduce a mutually invertible change of variable $t$ into the original equations and then find a solution to this new equation in the form with ordinary exponential, one can obtain a completely different Taylor expansion of the desired function. The essence of this method comes down to resuming the series.