| Th. Blesgen (Leipzig): A Multiscale Approach to Diffusion Induced Segregatio | |||||||||
| Diffusion Induced Segregation is a particular class of phenomena in mineralogy where the segregation only starts after the concentration of a diffusor penetrating the solid from outside exceeds a certain threshold. In a first step, based on a thermodynamical description, a system of partial differential equations is derived to model the process. The existence theory is shortly summarised. By ab initio methods, the actual free energies are approximately computed to perform high-precision finite-element simulations. To achieve this goal, the elastic constants of the single phases are computed and molecular dynamics computations are done to get the diffusion parameters. Also, the lattice dependence and different parts of the entropy are analysed. | |||||||||
| J. Dorignac (Boston): Band-edge mode bifurcation in partially isochronous Hamiltonian lattices | |||||||||
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We generalize the results obtained by S. Flach [S. Flach, Physica D 91
(1996) 223] regarding the bifurcation of band-edge modes (BEM) in
Hamitonian lattices to the case where these modes are partially
isochronous. We show that the bifurcation energy of a BEM is
intimately related to the low-energy behavior of ist frequency
(partial isochronism) and derive its explicit expression. In addition,
we show that the slow modulations of small-amplitude BEMs are governed
by a discrete nonlinear Schrodinger equation whose nonlinear exponent
is proportional to the degree of isochronism of the corresponding
orbit. We shall briefly discuss the link it provides between the
bifurcation of BEMs and the possible emergence of localized modes such
as discrete breathers.
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| S. Flach (Dresden): Localization in systems with nonlinear long-range interactions --- from discrete breathers to $q$-breathers | |||||||||
|
I will introduce the concept of discrete breathers - time-periodic
spatially localized excitations in nonlinear lattices.
The discussion of their properties in the presence of long range interactions
as well as purely nonlinear interactions will bring me to the
recent observation of q-breathers - time-periodic excitations
which are localized in the reciprocal q-space. These new excitations
are used to explain the main parts of the fifty years old Fermi-Pasta-Ulam
problem, when normal modes do not equilibrate.
I will derive analytical estimates of stability and delocalization thresholds for q-breathers
and discuss further potential developments in this field.
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| J. Giannoulis (Berlin): Three-wave interaction in discrete lattices | |||||||||
|
We consider the interaction of three pulses in a multidimensional, monoatomic
lattice. The scalar displacement of each atom is described by Newtons
equations of motion, and is due to the pairwise interaction of the atoms with
arbitrary many neighbours, as well as due to the embedding of the lattice in an
external field.
Modelling the pulses as macroscopic amplitude modulations of three plane waves
which are in resonance, we derive and justify a system of three nonlinearly
coupled equations which describe the macroscopic evolution of the amplitudes.
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| M. Herrmann (Berlin): On the modulation theory for the atomic chain | |||||||||
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| Sp. Kamvissis
(Leipzig): Semiclassical Limit of the Integrable Nonlinear Schrödinger Equation | |||||||||
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I will review recent results of several people concerning the
semiclassical (or small dispersion) limit of the (defocusing and focusing)
nonlinear Schrödinger equation with cubic nonlinearity.
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| O. Kastner
(Bochum): Atomistic simulation of an elastic-plastic body with shape memory | |||||||||
|
We investigates the thermodynamic properties of a qualitative
atomistic model of austenite-martensite
transitions as they occur in shape memory alloys. The model, still
in 2D, employs Lennard-Jones potentials for the determination of the
atomic interactions. By use of two atom species it is possible to
identify three different lattice structures in 2D, interpreted as
austenite and two variants of martensite. A test body consisting of 41
particles tends to transform uniformly between these
configurations. The free energy of this test body may be determined
by tensile tests. Due to temperature-dependent, non-monotone (load,
strain) characteristics, the test body exhibits non-convex isotherms
of the free energy. The thermodynamic criterion of phase equilibrium
is evaluated: Interestingly, austenite-martensite transitions of the
small test body obey this criterion entirely, including
temperature-dependent transition loads at the Maxwell lines, which
appear to be reversible in numerical experiments.
The model implies qualitatively the description of quasi-plasticity,
pseudo-elasticity and the shape memory effect. These processes become
visible in larger bodies which allow for the creation of micro
structures upon phase transition. MD simulations of a chain consisting
of 11 crystallites investigated formerly are presented, which are diagonally
linked. Temperature, strain or load of the entire chain may be
controlled in tensile tests.
The chain may be regarded as particular simple model of a larger
body. Quasi-plasticity appears as the result
of martensitic de-twinning of the chain upon loading at low
temperature and pseudo-elasticity appears as load-induced
austenite-martensite transitions at high temperature. The shape memory
effect turns out as temperature-induced martensite-austenite phase
transitions of the unloaded chain, just like in shape memory alloys.
The chain is motivated by works on elasto-plasticity employing
snap springs for the modeling of lattice transitions. Snap springs
are bi-stable mechanical devices, therefore transitions are always load induced
and imply hystereses. In a way the snap springs are replaced by
use of tri-stable, thermalized crystallites here. Hence the chain
represents qualitatively a thermo-mechanical material.
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| T.
Kriecherbauer (Bochum): Shocks and beyond --- the continuum limit of the Toda lattice | |||||||||
|
In this talk I will report on joint work with J. Baik, K. McLaughlin,
and P.D. Miller.
By choosing appropriate initial conditions the Toda lattice can be
viewed as a spatial discretisation of the nonlinear hyperbolic system
$$ A_t = 4 B B_x, \quad B_t = B A_x \, .$$
Solutions of this system may develop shocks in finite time, leading
to oscillations in the corresponding Toda flow on the microscopic
(i.e. lattice) scale. Using the fact that the Toda lattice belongs to
the select class of completely integrable systems we are able toxs
describe the dynamical behavior of the Toda lattice before and after
shock times. The proof will lead us via the inverse spectral theory of
Jacobi matrices to asymptotic results for polynomials orthogonal with
respect to discrete weights which have applications in Random Matrix
Theory as well.
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| M. Kunik (Magdeburg): Time-Frequency Analysis for the Atomic Chain | |||||||||
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| D. Levermore (Berlin): To be announced | |||||||||
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| F. Macia (Madrid): High-frequency wave propagation in discrete and continuous media | |||||||||
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We shall discuss two specific aspects of high-frequency wave
propagation: (1) the wave equation on a discrete torus, and (2) the
Schrödinger equation on compact manifolds. We shall focus on
understanding how the geometry of the classical underlying systems
determine the high-frequency behavior of waves. Concerning (1), our
analysis will rely on the construction and manipulation of discrete
Wigner functions; we shall give an application of this analysis to the
study of strong forms of unique continuation for semidiscrete wave
equations. Regarding (2), we shall give some results describing the
interplay between the geometry of the geodesic flow on a compact
manifold and the structure of Semiclassical/Wigner measures associated
to solutions to Schrödinger's equations.
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| A. Mielke (Berlin): Hamiltonian and Lagrangian formulation of modulation equations | |||||||||
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| S. Paleari (Milano): Metastability and dispersive shock waves in Fermi-Pasta-Ulam system | |||||||||
| We show the relevance of the dispersive analog of the shock waves,
described by Whitham equations, in the FPU dynamics. In particular we give
numerical evidence that metastable states in FPU are indeed constitued by
"Whitham trains" traveling through the chain, and that their long time
stability is related to the integrable nature of the underlyng continuum
approximation, i.e. the KdV equation. We also investigate and explain the
apparently elastic nature of the interaction between Whitham trains.
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| C. Patz
(Stuttgart/Paris): Dispersive behavior in harmonic oscillator chains | |||||||||
|
We study the long-time dynamics of a one-dimensional infinite chain of
particles linked by nearest- and next-nearest-neighbour harmonic springs. In
particular, the dispersion of energy is analysed. Given compactly supported
initial conditions, the energy distribution is explained using numerical
simulations and decay rates for the displacements and velocities are proved
using methods for oscillatory integrals.
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| J. Rademacher (Berlin): Geometry of travelling waves in Riemann problems for the hyperbolic limit of atomic chains | |||||||||
|
In several cases the macroscopic hyperbolic limit dynamics of atomic
chains can be described by modulated travelling waves. Motivated
by numerical experiments of Riemann problems, we discuss the arising
macroscopic structures in terms of the geometry of the underlying travelling
waves.
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| H. Uecker (Karlsruhe): A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom | |||||||||
|
The spatially periodic Kuramoto--Sivashinsky equation (pKS)
$$
\partial_t u=-\partial_x4 u-c_2\partial_x2 u+2\delta\partial_x(\cos( x)u)
-\partial_x(u2), \quad u(t,x)\in{\mathbb R},\ t\geq 0,\ x\in{\mathbb R},
$$
can be considered as a model problem for the flow of a viscous liquid film
down an inclined wavy plane.
For given $c_2\in{\mathbb R}$ and $\delta\geq 0$ it has a one dimensional
family of spatially periodic stationary solutions
$u_s(\cdot;c_2,\delta,u_0)$,
parametrized by the mass $u_0=\frac 1 {2\pi}\int_0^{2\pi} u_s(x) \,{\rm d}x$.
Depending on the parameters these stationary solutions
can be linearly stable or unstable, with a long wave instability.
Using Bloch wave analysis we separate the long scale from the
short scale coming from the bottom profile. Then, using renormalization
group methods, we show that in the stable case localized perturbations
decay with a polynomial rate and in a universal self-similar way: the limiting profile is determined by a Burgers equation in Bloch wave space.
We also discuss wave patterns in the linearly unstable case.
Joint work with Andreas Wierschem, Bayreuth
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| A. Vainchtein (Pittsburgh): Kinetics of a phase boundary: Lattice model and quasicontinuum approximation | |||||||||
|
Martensitic phase transitions are often modeled by mixed type
hyperbolic-elliptic systems. Such systems lead to ill-posed initial-value
problems unless they are supplemented by an additional kinetic relation.
In this talk I will discuss how one can explicitly compute an appropriate
closing relation by replacing continuum model with its natural discrete
prototype. A moving phase boundary is
represented by a traveling wave solution of a fully inertial discrete
model for a bi-stable lattice with harmonic long-range interactions.
Although the microscopic model is Hamiltonian,
it generates macroscopic dissipation which can be specified
in the form of a relation between the velocity of the discontinuity
and the conjugate configurational force.
The dissipation at the macrolevel is due
to the induced radiation of lattice waves carrying
energy away from the propagating front. For sufficiently fast phase
boundaries the kinetic relation predicted by the discrete model can be
captured by a dispersive quasicontinuum approximation that includes
non-classical corrections to both potential and kinetic energies.
This is a joint work with Lev Truskinovsky, Ecole Polytechnique.
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| St. Venakides (Durham): Focusing Nonlinear Schrödinger equation: Rigorous Semiclassical Asymptotics | |||||||||
|
The NLS equation describes solitonic transmission in fiber optic
communication and is generically encountered in propagation through
nonlinear media. One of its most important aspects is its modulational
instability: regular wavetrains are unstable to modulation and break
up to more complicated structures.
The IVP for the NLS equation is solvable by the method of inverse
scattering. The initial spectra data of the Zakharov Shabhat (ZS)
operator, a particular linear operator having the solution to NLS as
the potential, are calculated from the initial data of the NLS; they
evolve in a simple way as a result of the integrability of the
problem, and produce the solution to NLS through the inverse spectral
transformation.
In collaboration with A. Tovbis, we have developed a one
parameter family of initial data for which the derivation of the
spectral data is explicit.
Then, in collaboration with A. Tovbis and
X. Zhou, we have obtained the follwing results:
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| J. Zimmer (Bath): Travelling Waves for Nonconvex FPU Lattices | |||||||||
|
Travelling waves in a one-dimensional chain of atoms will be
investigated. The aim is to allow for nonconvex energy densities,
which occur in the theory of phase transforming solids, such as
martensitic crystals. The existence of solitary waves with a
prescribed asymptotic strain will be shown under certain assumptions
on the asymptotic strain and the wave speed. Connections to previous
results will be discussed.
This is joint work with Hartmut Schwetlick (Bath).
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