Filters

2006 - 20084

More filters

20194
Reset filters

Settings

search.filters.applied.f.access: openAccess × Author: Herrmann, Michael × End date: 2009 × Start date: 2006 × Type: Text ×

Search Results

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    Wave trains, solitons and modulation theory in FPU chains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Dreyer, Wolfgang; Herrmann, Michael; Rademacher, Jens D.M.
    We present an overview of recent results concerning wave trains, solitons and their modulation in FPU chains. We take a thermodynamic perspective and use hyperbolic scaling of particle index and time in order to pass to a macroscopic continuum limit. While strong convergence yields the well-known p-system of mass and momentum conservation, we generally obtain a weak form of it in terms of Young measures. The modulation approach accounts for microscopic oscillations, which we interpret as temperature, causing convergence only in a weak, average sense. We present the arising Whitham modulation equations in a thermodynamic form, as well as analytic and numerical tools for the resolution of the modulated wave trains. As a prototype for the occurrence of temperature from oscillation-free initial data, we discuss various Riemann problems, and the arising dispersive shock fans, which replace Lax-shocks. We predict scaling and jump conditions assuming a generic soliton at the shock front.
  • Thumbnail Image
    Item
    Infinite harmonic chain with heavy mass
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Herrmann, Michael; Segatti, Antonio
    Modelling a crystal with impurities we study an atomic chain of point masses with linear nearest neighbour interactions. We assume that the masses of the particles are normalised to $1$, except for one heavy particle which has mass $M$. We investigate the macroscopic behaviour of such a system when $M$ is large, and time and space are scaled accordingly. As main result we derive a PDE for the light particles that is coupled with an ODE for the heavy particle.
  • Thumbnail Image
    Item
    Continuum descriptions for the dynamics in discrete lattices : derivation and justification
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Giannoulis, Johannes; Herrmann, Michael; Mielke, Alexander
    The passage from microscopic systems to macroscopic ones is studied by starting from spatially discrete lattice systems and deriving several continuum limits. The lattice system is an infinite-dimensional Hamiltonian system displaying a variety of different dynamical behavior. Depending on the initial conditions one sees quite different behavior like macroscopic elastic deformations associated with acoustic waves or like propagation of optical pulses. We show how on a formal level different macroscopic systems can be derived such as the Korteweg-de Vries equation, the nonlinear Schroedinger equation, Whitham's modulation equation, the three-wave interaction model, or the energy transport equation using the Wigner measure. We also address the question how the microscopic Hamiltonian and the Lagrangian structures transfer to similar structures on the macroscopic level. Finally we discuss rigorous analytical convergence results of the microscopic system to the macroscopic one by either weak-convergence methods or by quantitative error bounds.
  • Thumbnail Image
    Item
    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.