Modulated Fourier expansions for continuous and discrete oscillatory systems

This talk reviews some of the phenomena and theoretical results on the long-time energy behaviour of continuous and discretized oscillatory systems that can be explained by modulated Fourier expansions: long-time preservation of total and oscillatory energies in oscillatory Hamiltonian systems and their numerical discretisations, near-conservation of energy and angular momentum of symmetric multistep methods for celestial mechanics, metastable energy strata in nonlinear wave equations, and long-time stability of plane wave solutions of nonlinear Schroedinger equations.

We describe what modulated Fourier expansions are and what they are good for. Much of the presented work was done in collaboration with Ernst Hairer. Some of the results on modulated Fourier expansions were obtained jointly with David Cohen and Ludwig Gauckler.