Workshop on

Macroscopic Limits and Modulation Theory

for PDEs and Lattice Models

List of Abstracts

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
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.

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.

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.

M. Herrmann (Berlin):
On the modulation theory for the atomic chain

Sp. Kamvissis (Leipzig):
Semiclassical Limit of the Integrable Nonlinear Schrödinger Equation
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.

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.

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.

M. Kunik (Magdeburg):
Time-Frequency Analysis for the Atomic Chain

D. Levermore (Berlin):
To be announced

F. Macia (Madrid):
High-frequency wave propagation in discrete and continuous media
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.

A. Mielke (Berlin):
Hamiltonian and Lagrangian formulation of modulation equations

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.

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.

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.

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

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.

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:

  1. We prove the existence and basic properties of the first breaking curve (curve in space-time above which the character of the solution changes by the emergence of a new oscillatory phase) and show that for pure radiation no further breaks occur.
  2. We construct the solution beyond the first break-time.
  3. We derive a rigorous estimate of the error.
  4. We derive rigorous asymptotics for the large time behaviour of the system in the pure radiation case.

Finally, in collaboration with R. Buckingham, we solve the so called "shock problem", that describes the long time evolution of two plane waves left and right of the origin respectively colliding at the origin.

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).