Exponential averaging under rapid quasiperiodic forcing

Abstract

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.