Modulated Fourier expansions of highly oscillatory
differential equations
David Cohen, Ernst Hairer and Christian Lubich
Abstract.
Modulated Fourier expansions are developed as a tool for gaining
insight into the long-time behaviour of Hamiltonian systems with
highly oscillatory solutions.
Particle systems of Fermi-Pasta-Ulam type
with light and heavy masses are considered as an example.
It is shown that the harmonic energy of the highly oscillatory
part is nearly conserved over times that are exponentially long
in the high frequency. Unlike previous approaches to such
problems, the technique used here does not employ nonlinear coordinate
transforms and can therefore be extended to the analysis of
numerical discretizations.
Key Words.
Modulated Fourier expansion, Fermi-Past-Ulam problem,
conservation of energy, highly oscillatory differential equations.