Macroscopic dynamics in microscopic nonlinear oscillator chains
Johannes Giannoulis, Alexander Mielke · PAMM · 2005
Abstract We consider an exactly periodic oscillator chain as a microscopic model for a one‐dimensional continuum. In the linearized model there exist space‐time harmonic solutions. Our aim is to investigate the macroscopic modulation of these microscopic patterns in the regime of small amplitudes. To this end, depending on the fixed wave number and frequency, we derive the associated macroscopic continuum model under suitable non‐resonance conditions. For non‐zero wave numbers the macroscopic model consists of a system of nonlinear Schrödinger equations. Moreover, we justify this formal derivation by showing that the approximation through the macroscopic model stays in the proximity of the solution of the oscillator chain over time intervals with a positive macroscopic length. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)