Dispersive evolution of pulses in oscillator chains with general interaction potentials
- Giannoulis, Johannes
- Mielke, Alexander
2010 Mathematics Subject Classification
- 34E13 34C20 37K60 70F45 70K70
- Nonlinear oscillator chain, multiscale ansatz, nonlinear Schroedinger equation, justification of modulation equations, normal form transformation, nonresonance conditions
We consider the dispersive evolution of a single pulse in a nonlinear oscillator chain embedded in a background field. We assume that each atom of the chain interacts pairwise with an arbitrary but finite number of neighbours. The pulse is modeled as a macroscopic modulation of the exact spatiotemporally periodic solutions of the linearized model. The scaling of amplitude, space and time is chosen in such a way that we can describe how the envelope changes in time due to dispersive effects. By this multiscale ansatz we find that the macroscopic evolution of the amplitude is given by the nonlinear Schroedinger equation. The main part of the work is focused on the justification of the formally derived equation: We show that solutions which have initially the form of the assumed ansatz preserve this form over time-intervals with a positive macroscopic length. The proof is based on a normal form transformation constructed in Fourier space, and the results depend on the validity of suitable nonresonance conditions.
- Discrete Contin. Dyn. Syst. Ser. B, 6 (2006) pp. 493--523.