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Author: Giannoulis, Johannes × Author: Mielke, Alexander × End date: 2009 × Start date: 2008 × Type: Text ×

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    High-frequency averaging in semi-classical Hartree-type equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Giannoulis, Johannes; Mielke, Alexander; Sparber, Christof
    We investigate the asymptotic behavior of solutions to semi-classical Schröodinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.
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    Lagrangian and Hamiltonian two-scale reduction
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Giannoulis, Johannes; Herrmann, Michael; Mielke, Alexander
    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.