Advances in Differential Equations

Exponential averaging under rapid quasiperiodic forcing

Karsten Matthies

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We derive estimates on the magnitude of the interaction between a wide class of analytic partial differential equations and a high-frequency quasiperiodic oscillator. Assuming high regularity of initial conditions, the equations are transformed to an uncoupled system of an infinite dimensional dynamical system and a linear quasiperiodic flow on a torus; up to coupling terms which are exponentially small in the smallest frequency of the oscillator. The main technique is based on a careful balance of similar results for ordinary differential equations by Simó, Galerkin approximations and high regularity of the initial conditions. Similar finite order estimates assuming less regularity are also provided. Examples include reaction-diffusion and non-linear Schrödinger equations.

Article information

Adv. Differential Equations, Volume 13, Number 5-6 (2008), 427-456.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C29: Averaging method
Secondary: 34G20: Nonlinear equations [See also 47Hxx, 47Jxx] 35A35: Theoretical approximation to solutions {For numerical analysis, see 65Mxx, 65Nxx} 35K57: Reaction-diffusion equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol d diffusion


Matthies, Karsten. Exponential averaging under rapid quasiperiodic forcing. Adv. Differential Equations 13 (2008), no. 5-6, 427--456.

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