We consider the nonlinear model of an infinite oscillator chain embedded in a background field. We start from an appropriate modulation ansatz of the space-time periodic solutions to the linearized (microscopic) model and derive formally the associated (macroscopic) modulation equation, which turns out to be the nonlinear Schrödinger equation. Then we justify this necessary condition rigorously for the case of nonlinearities with cubic leading terms; i.e. we show that solutions that have the form of the assumed ansatz for t = 0 preserve this form over time-intervals with a positive macroscopic length. Finally, we transfer this result to the analogous case of a finite but large periodic chain and illustrate it by a numerical example.

In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential U , which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness, and stability results for the flow in the space of measures induced by the continuity equation.

We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic interaction of these pulses.

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 Schrödinger 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.

We present a rigorous derivation of classical molecular dynamics (MD) from quantum molecular dynamics (QMD) that applies to the standard Hamiltonians of molecular physics with Coulomb interactions. The derivation is valid away from possible electronic eigenvalue crossings.