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We study the dispersive evolution of modulated pulses in a  nonlinear oscillator chain embedded in a background field. The atoms  of the chain interact pairwise with an arbitrary but finite number of  neighbors. The pulses are modeled as macroscopic modulations 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 Schrodinger 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.

Dispersive evolution of pulses in oscillator chains with general interaction potentials