# Continuous changes of variables and the Magnus expansion

## TL;DR

This paper develops a framework for Magnus and Floquet-Magnus expansions applicable to nonlinear differential equations, introducing continuous variable transformations and demonstrating their use on specific nonlinear systems.

## Contribution

It presents a novel formulation of Magnus and Floquet-Magnus expansions for nonlinear equations using continuous variable transformations.

## Key findings

- Explicit first terms of Floquet-Magnus expansion for Van der Pol oscillator
- Explicit first terms of Floquet-Magnus expansion for nonlinear Schrödinger equation
- New methodology for nonlinear differential equations

## Abstract

In this paper, we are concerned with a formulation of Magnus and Floquet-Magnus expansions for general nonlinear differential equations. To this aim, we introduce suitable continuous variable transformations generated by operators. As an application of the simple formulas so-obtained, we explicitly compute the first terms of the Floquet-Magnus expansion for the Van der Pol oscillator and the nonlinear Schr\"odinger equation on the torus.