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 Schrödinger equation on the torus.
Highlights
The Magnus expansion constitutes nowadays a standard tool for obtaining both analytic and numerical approximations to the solutions of non-autonomous linear differential equations
We review several procedures to derive the Magnus expansion for the linear equation (1) in section 2 and introduce a binary operator that will play an important role in the sequel
There, we show how they reproduce the classical expansions for linear differential equations
Summary
The Magnus expansion constitutes nowadays a standard tool for obtaining both analytic and numerical approximations to the solutions of non-autonomous linear differential equations. The operator Φt obeys a linear differential equation which is formally solved with the corresponding Magnus expansion [7].
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