Project within the DFG Priority Program SPP 1095
Analysis, Modeling and Simulation of Multiscale Problems
Macroscopic dynamics in discrete lattices
- Project Leader
Prof. Dr. Alexander Mielke - Researcher
Dr. Johannes Giannoulis, Carsten Patz (on leave to Paris) - Duration (of third period)
August 2004 - July 2006 - Cooperation
Wolfgang Dreyer, Michael Herrmann, Jens Rademacher, Christoph Sparber - General Working Field
Applied Mathematics. Analysis of differential equations. Connection between partial differential equations and spatially discrete systems. Continuum mechanics. - Summary
We investigate the dynamic behavior of atomistic models for crystals which are given by infinitely many coupled ordinary differential equations. The aim is to justify macroscopic continuum limits for such microscopic models. For this purpose we prove that certain solution classes for the atomistic model can be described by macroscopic partial differential equations if we rescale the system such that the atomic distance tends to 0 and the time scaling is chosen appropriately.In the harmonic, linear case we study the interaction of the dispersive energy transport with effects due to interfaces, surfaces or other localized structures. We expect to derive a coupled system consisting of the energy-transport equation for the Wigner measure in the bulk and a new equation on the interface which is coupled to the bulk by reflection and transmission laws.
In the weakly nonlinear setting we will study the interaction of several pulses modulating different carrier frequencies under resonance and nonresonance conditions. Finally, we consider stability questions for large amplitude, periodic traveling waves. Moreover, we investigate in what sense Whitham's equation for modulations of traveling waves is a rigorous macroscopic limit.
| teaching | projects | conferences | publications | short CV | miscellaneous |