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.

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Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system.In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions.In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.

We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and non-resonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.