Mathematics
of Materials and Macromolecules: Multiple Scales, Disorder,
and Singularities, September 2004  June 2005
Abstracts:
IMA
Workshop:
October
2529, 2004
Photo
Gallery Materials
from Talks
Mark
J. Ablowitz
(Department of Applied Mathematics, University of Colorado)
mark.ablowitz@colorado.edu
Wave
Collapse in Nonlocal Nonlinear Schrödinger Equations
Wave
collapse occurs in nonlinear media whose governing equations
have quadratic nonlinearities. Examples include water waves
and nonlinear optics. Although these two physical problem
are very different, the equations that govern their dynamics
are similar. The equations, which couple the first harmonic
to the mean terms in a quasimonochromatic amplitude perturbation
expansion are nonlocal nonlinear Schrodinger systems. They
are sometimes referred to as BenneyRoskes or DaveyStewartson
type. The two dimensional ground state solution, is found
to play an important role in the wave collapse mechanism.
Stan
Alama (Department of Mathematics and Statistics,
McMaster University) alama@univmail.cis.mcmaster.ca
Vortices
and Pinning Effects for the GinzburgLandau Model in Multiply
Connected Domains
Slides: pdf
We
consider the twodimensional GinzburgLandau model with magnetic
field, for a superconductor with multiply connected crosssection.
We study energy minimizers in the London limit as the GinzburgLandau
parameter tends to infinity, to determine the number and asymptotic
location of vortices. We show that the holes act as pinning
sites, acquiring nonzero winding for bounded fields and attracting
all vortices away from the interior for fields up to a critical
value. At the critical level the pinning effect breaks down,
and vortices appear in the interior of the superconductor at
locations which we identify explicitly, in terms of the solutions
of an elliptic boundary value problem. The method involves sharp
upper and lower energy estimates, and a careful analysis of
the limiting problem which captures the interaction between
the vortices and the holes.
Suman
Anand (Polymeric and Soft Material Division,
National Physical Laboratory) sanand@mail.nplindia.ernet.in
Investigation
of the Spectral Properties at Phase Singularity (poster)
Joint
work with M.N.Kamalasanan.
Recently
optical vortices have attracted considerable attentions because
of the phase singularity and the characteristic intensity distribution.
In particular, the dark region is very useful to trap and guide
atoms. Several methods of generating the optical vortices have
been reported so far.
We have demonstrated a simple generation of optical vortices
produced by computergenerated holograms near the focus of a
converging spatially fully coherent polychromatic wave using
a simple experimental technique. The anomalous behavior of the
spectra at and in the vicinity of phase singularity has also
been studied. It is found that the spectrum of the wave on the
hologram, consisting of a single spectral profile splits into
two peaks at the onaxis point and shows red shift and blue
shift around the phase singularity. These spectacular spectral
changes also take place in the vicinity of dark rings of Airy
pattern formed with spatially coherent, polychromatic light
diffracted at a circular aperture. In another experiment, it
is also demonstrated that spectral degree of coherence of partially
coherent light may posses isolated points at which its phase
is singular, and that in the neighborhood of these points the
phase may possess a vortex structure. Here, also we found that
at the point of phase singularity the spectrum splits into two
peaks.
Fabrice
Bethuel (Laboratoire JacquesLouis Lions, ENPCCERMA
) bethuel@ann.jussieu.fr
Slow
Motion Collisions and Phase Vortex Interaction in the TwoDimensional
Parabolic GinzburgLandau Dynamics
The talk is based on a joint work with D.Smets and G. Orlandi,
in which we describe a natural framework for the vortex dynamics
the parabolic complexGinzburglandau equation on the plane.
This general setting does not rely on any assumption of wellpreparedness
and has the advantage to be valid even after possible collision
times. We analyze carefully collisions leading to annihilation,
and identify a new phenomenon, the phasevortex interaction,
related to persistency of low frequency oscillations, and leading
to an unexpected drift in the motion of vortices.
I
will also discuss open issues, as the splitting of multiple
degree vortices,, collisions without annihilation, ...
Rustum
Choksi (Department of Mathematics, Simon Fraser
University) choksi@cs.sfu.ca
Phase
Separation in Block Copolymer Systems: Paradigms for Multiscale
Phase Separation
Slides: pdf
Systems
of block copolymers are paradigms for phase separation on several
scales and display a wide range of geometric structures. Their
remarkable ability for selfassembly (in the melt phase) into
various ordered structures aids in the construction of many
so called "designer materials"; Indeed, these geometrical structures
are key to the many properties that make diblock copolymers
of great technological interest.
This
talk will be devoted to certain macroscopic models for phase
separation in diblock copolymer melts and homopolymer blends,
their derivation, and the mathematical questions that arise.
In particular, within the context of these models I will discuss
both the periodicity and scale of these patterns, and their
geometry. The latter appears to be closely related to the periodic
isoperimetric problem which, in three dimensions, remains an
open problem in classical geometry.
Parts
of the talk will cover joint work with X.
Ren (Utah State University), joint work with G.
Alberti (University of Pisa) and F.
Otto (University of Bonn), and work in progress with
P. Sternberg (Indiana University).
Alan
Dorsey (Department of Physics, University of
Florida) dorsey@phys.ufl.edu
Electronic
"Liquid Crystals"
There
is growing experimental evidence that electrons confined to
two dimensions (in a semiconductor heterostructure, for instance)
at low temperatures and high magnetic fields can display a plethora
of partially ordered phases which have the same symmetries as
classical liquid crystal phases, such as nematics and smectics.
I will review the experimental evidence for these novel quantum
phases of matter, and discuss their phenomenology, with an emphasis
on the key role of topological defects (dislocations) partially
restoring broken symmetries.
Slides
from a related talk
 December 2003 colloquium at NYU: html
pdf
ps
ppt
Andre
Geim
(Centre for Mesoscience and Nanotechnology, University of Manchester)
geim@man.ac.uk
SubAtomic
Movements of Domain Walls in CrystalLattice Potential
Slides:
pdf
The discrete nature of the crystal lattice bears on virtually
every material property but it is only when the size of condensedmatter
objects  e.g., dislocations, vortices in superconductors,
domain walls in magnetic materials  becomes comparable to the
lattice period, that the discreteness reveals itself explicitly.
The associated phenomena are usually described in terms of the
Peierls (?atomic washboard?) potential, which was first introduced
for the case of dislocations at the dawn of the condensedmatter
era. Since then, it has been invoked in many situations to explain
certain features in bulk properties of materials but never observed
directly. We have succeeded for the first time to monitor experimentally
how a single domain wall moves through individual peaks and
troughs of the atomic landscape. The wall becomes trapped between
adjacent crystalline planes, which results in its propagation
by distinct jumps matching the periodicity of the Peierls potential
(Nature 426, 812, 2003). As a domain wall moves from one Peierls
valley to another, it becomes unexpectedly flexible at Peierls
ridges, which we attribute to atomicsize kinks propagating
along a wall in the transient bistable position. The physics
of topological defects at this true atomic scale is badly understood
and seems to require a theory beyond the existing models.
Tiziana
Giorgi (Department
of Mathematical Sciences, New Mexico State University) tgiorgi@nmsu.edu
Superconductors
Surrounded by Normal Materials (poster)
We
consider a generalized GinzburgLandau energy functional modeling
a superconductor surrounded by a material in the normal state.
In this model the order parameter is defined in the whole space.
We derive existence of a global minimizer in weighted Sobolev
spaces for both square integrable and constant applied magnetic
fields. We then prove boundedness and classical elliptic estimates
for the oder parameter, in order to study the loss of superconductivity
for high applied magnetic fields. In twodimensions for the
general case and in threedimensions for the case of constant
permeability, we show the existence of an upper critical field
above which the only finite energy weak solutions are the normal
states. For the threedimensional case, we show that as the
applied field tends to infinity finite energy weak solutions
tend to the normal state.
Boaz
Ilan (Department of Applied Mathematics, University
of Colorado at Boulder) boaz@colorado.edu
Singularity
Formation in Nonlinear Schrodinger Equations with FourthOrder
Dispersion
Slides: pdf
Similarly
to the classical NLS, NLS equations with fourthorder dispersion
can admit singularity formation. Such equations arise in the
studies of opticalbeam propagation in fiber arrays and in the
numerical analysis of finitedifference discretizations of the
NLS. We extend the classicalNLS globalexistence theory to
the NLS with isotropic and anisotropic fourthorder dispersion.
Using the results on the critical exponent, it is shown how
fourthorder dispersion can assist in the physical realization
of spatiotemporal "light bullets" in a pure Kerr medium. These
are recent works with Gadi Fibich, George Papanicolaou, Steve
Schochet, and Shimshon BarAd.
Sookyung
Joo (IMA Postdoc) sjoo
at ima.umn.edu http://www.ima.umn.edu/~sjoo/
The
Phase Transitions Between the Chiral Nematic and Smectic Liquid
Crystals (poster)
We study the ChenLubensky model to investigate the phase transition
between chiral nematic and smectic liquid crystals. First, we
prove the existence of the minimizers in an admissible set where
the order parameter vanishes on the boundary. The splay, twist,
and bend Frank constants are considered to diverge near N* 
C* phase transition based on physical observation, while only
twist and bend constants can be assumed to diverge near N* 
A* transition. Then we describe the transition temperatures
for both smectic A side and smectic C side when a domain is
a considerably large liquid crystal region confined in two plates.
Robert
V. Kohn
(Department of Mathematics, Courant Institute) kohn@courant.nyu.edu
A
DeterministicControlBased Approach to Interface Motion
Slides:
pdf
Paper: pdf
Interface
motion is central to many application areas, including materials
science and computer vision. Motion by curvature is a basic
example, in which the normal velocity of the interface is equal
to its curvature. The levelset approach, now 15 years old,
represents the evolving interface as the zerolevelset of an
evolving PDE. This viewpoint has been extremely successful for
both simulation and analysis.
I'll
discuss joint work with Sylvia Serfaty, which develops a new
perspective on the levelset approach to motion by curvature
and related interface motion laws. We show, loosely speaking,
that the levelset PDE is the value function of a deterministic
twoperson game. More precisely, we give a family of discretetime,
twoperson games whose value functions converge in the continuoustime
limit to the solution of the motionbycurvature PDE. This result
is unexpected, because the value function of a deterministic
control problem is normally a firstorder HamiltonJacobi equation,
while the levelset formulation of motion by curvature is a
secondorder parabolic equation.
Richard
Kollar
(IMA Postdoc) kollar@ima.umn.edu
http://www.ima.umn.edu/~kollar/
Linear
Stability of Vortices in BoseEinstein Condensates
(poster)
The
existence of localized vortices in BoseEinstein condensates
was experminetally and analytically confirmed in the previous
years. Although there is a large literature on their linear
stability, the rigorous and complete approach to this problem
is absent. We study the simplest case of a single localized
vortex trapped in an harmonic trap in the twodimensional approximation.
We use the Evans function method which proved to be a robust
and reliable technique for studying nonlinear eigenvalue problems.
We confirm that singlyquantized vortices are linearly stable
and that the linear stability of multiquantized vortices depends
on the diluteness of a condensate, with alternating intervals
of stability and instability. Moreover, we propose a significant
reduction of the numerical cost of the algorithm by replacing
the traditional winding number calculation by using the information
on the Krein signature of possible unstable eigenvalues. (This
is a joint work with Robert L. Pego.)
Matthias
Kurzke (IMA Postdoc) kurzke@ima.umn.edu
http://www.ima.umn.edu/~kurzke/
Boundary
Vortices in Thin Magnetic Films (poster)
In
a certain thinfilm limit, the micromagnetic functional reduces
to a competition between a Dirichlet term and a boundary term
penalizing nontangential magnetization. We show that in the
limit where the boundary term is strongly penalized, minimizing
magnetizations develop two halfvortices at the boundary. Quite
similar to results for GinzburgLandau interior vortices, we
can show that the singular part of the energy depends only on
the number of vortices, while their interaction is encoded in
the next order term, a renormalized energy like that of BethuelBrezisHelein
for GinzburgLandau.
Robert
P. Lipton
(Department of Mathematics, Louisiana State University) lipton@math.lsu.edu
Optimal
Lower Bounds on the ElectricField Concentration in Random Dielectric
Media (poster)
The
study of failure initiation in heterogeneous random dielectrics
requires one to assess the effect of the local electric field
concentrations generated by macroscopic potential gradients.
Macroscopic quantities sensitive to the local field behavior
include higher order moments of the electric field inside the
composite. In this context the effective dielectric constant
can be viewed as a weighted first moment of the electric field.
The
results presented here concern twophase dielectric composites
and provide optimal lower bounds on all higher moments of the
electric field. The bounds depend upon the statistics of the
random media gathered from image analysis. The moments can be
of infinite order. An optimal lower bound on the essential supremum
of the magnitude of the electric field is presented that depends
explicitly on the twopoint statistics of the random composite.
Alexander
A. Nepomnyashchy
(Department of Mathematics, Technion  Israel Institute of Technology)
nepom@techunix.technion.ac.il
Feedback
Control of Singularities in Models of Directional Solidification
Joint
work with V. Gubareva (Technion),
A.A. Golovin (Northwestern University),
and V. Panfilov (University of
Nevada, Reno)
The
nonlinear evolution of the longwave morphological instability
in a directional solidification front characterized by a small
segregation coefficient in some cases does not saturate, leading
to the formation of "deep cells". In the framework of the Sivashinsky
equation, this effect manifests itself through a subcritical
instability of a plane solidification front and the development
of a singularity in a finite time ("blowup"). We investigate
the possibility of the suppression of the subcritical instability
and the blowup behavior by means of a feedback adjustment of
the pulling speed or temperature gradient. We show that the
feedback control eliminates the appearance of deep cells and
leads to the formation of localized structures at the crystalmelt
interface.
Also,
we investigate the possibility of the elimination of singularities
in the nonlinear development of a subcritical pulsatile instability
which may appear in the rapid solidification. In the framework
of the subcritical complex GinzburgLandau equation, we find
that the feedback control may lead to the formation of solitary
waves and multipulse waves.
Peter
PalffyMuhoray (Liquid Crystal Institute, Kent
State University) mpalffy@cpip.kent.edu
Photonic
Band Gap Aspects of Liquid Crystals
Liquid
crystals with modulated ground states are periodic dielectric
structures, and hence they are selfassembled photonic band
gap materials. Fluorescent emission is suppressed in the stop
band, but is enhanced at the band edges, and, above a pump threshold,
distributed feedback lasing occurs. We interpret experimental
observations of reflectance, fluorescence and lasing in terms
of the density of states, and give a detailed description of
this for helical cholesteric liquid crystals. Lasing is due
to the singularity in the density of states at the band edges.
Xingbin
Pan
(Department of Mathematics, East China Normal University) amaxbpan@zju.edu.cn
Nucleation
of Smectics
Similarity
of GinzburgLandau model of superconductivity and LandaudeGennes
model for liquid crystals was discovered by P. G. de Gennes,
and has been investigated by many physicists and mathematics.
In this talk we shall discuss possible similarity in the mathematical
theory of nucleation of superconductivity and that of smectics.
We shall also discuss some recent progress on the estimates
of the critical number Qc3 for liquid crystals, that is analogy
of the critical field Hc3 for superconductivity.
Francois
Peeters (Departement Fysica, Universiteit Antwerpen
(Campus Drie eiken), Universiteitsplein 1, B2610 Antwerpen)
francois.peeters@ua.ac.be
http://www.cmt.ua.ac.be/
Vortex
Matter in Nanostructured and Hybrid Superconductors
Slides:
html
pdf
ps
ppt
The
interplay between superconductivity and the inhomogeneous magnetic
field generated by nanostructured ferromagnets leads to new
vortex arrangements not found in homogeneous superconductors.
We consider two situations: 1) a ferromagnetic disk on top of
a thin superconducting film, and 2) a lattice of such ferromagnetic
disks separated by a thin oxide layer on top of a thin superconducting
film.
For
a single ferromagnetic disk magnetized perpendicular to the
plane of the superconducting film we found that antivortices
are stabilized in shells around a central core of vortices (or
a giant vortex) with size/magnetizationcontrolled `magic numbers.'
The transition between the different vortex phases occurs through
the creation of a vortexantivortex pair under the edge of the
magnetic disk. In the case of a lattice of ferromagnetic disks,
the antivortices form a rich spectrum of lattice states. In
the ground state the antivortices are arranged in the socalled
matching configurations between the ferromagnetic disks while
the vortices are pinned to the ferromagnets. The exact (anti)vortex
structure depends on the size, thickness and magnetization of
the magnetic dots, periodicity of the ferromagnetic lattice
and properties of the superconductor expressed through the effective
GinzburgLandau parameter *.
The experimental implications of our results such as magneticfieldinduced
superconductivity will be discussed.
The
theoretical analysis is based on a numerical .exact. solution
of the phenomenological GinzburgLandau equations.
Dmitry
Pelinovsky (Department of Mathematics, McMaster
University, Canada) dmpeli@math.mcmaster.ca
http://dmpeli.math.mcmaster.ca/
Discrete
Solitons and Vortices in Nonlinear Schrödinger Lattices
Preprint 1: pdf
Preprint
2:
pdf
We
consider the discrete solitons and vortices bifurcating from
the anticontinuum limit of the discrete nonlinear Schrodinger
(NLS) lattice. The discrete soliton in the anticontinuum limit
represents an arbitrary finite superposition of inphase or
antiphase excited nodes, separated by an arbitrary sequence
of empty nodes. The discrete vortices represent a finite set
of excited nodes with non
phase shifts between the adjacent nodes.
By
using stability analysis, we prove that the discrete solitons
are all unstable near the anticontinuum limit, except for the
solitons, which consist of alternating antiphase excited nodes.
We classify analytically and confirm numerically the number
of unstable eigenvalues associated with each family of the discrete
solitons.
By
using LyapunovSchmidt reductions, we study existence and stability
of symmetric, supersymmetric and asymmetric discrete vortices
in simplyconnected discrete contours on the plane.
Peter
Philip (IMA Postdoc) philip@ima.umn.edu
http://www.ima.umn.edu/~philip/
Optimal
Contol of a PDE with Weakly Singular Integral Operator: Tuning
Temperature Fields During Crystal Growth (poster)
Slides: pdf
We
consider the optimal control of the temperature gradient during
the modeling of diffusegray conductiveradiative heat transfer.
The problem arises from the aim to optimize crystal growth by
the physical vapor transport (PVT) method. The temperature is
the solution of a semilinear PDE where radiative heat trasfer
between cavity surfaces is modelled by a weakly singular integral
operator. Based on a minimum principle for the semilinear equation,
we establish the existence of an optimal solution as well as
necessary optimality conditions. The theoretical results are
illustrated by results of numerical computations.
Daniel
Phillips
(Department of Mathematics, Purdue University) phillips@math.purdue.edu
Existence
of Classical Solutions of the TimeDependent GinzburgLandau
Equations for a Bounded Superconducting Domain in a Vacuum
Joint
work with Patricia Bauman and Hala
Jadallah.
The
initial value problem for the GinzburgLandau equations modeling
currents in a threedimensional body surrounded by a vacuum
is a mixed, ellipticparabolic system. We prove existence,
uniqueness, and regularity results near the body's surface for
solutions. Moreover we compare solutions from different gauges.
Yves
Pomeau
(Department of Mathematics, University of Arizona) yves.pomeau@lps.ens.fr
Finite
Time Singularities of Solutions of the Euler Fluid Equations
in 3D
Joint
work with D. Sciamarella.
The
smoothness at all times of the solutions of the equations for
inviscid incompressible fluids in 3D with finite energy remains
a major unanswered question in applied mathematics. Long ago
Leray suggested to look at the equations for selfsimilar singularities
for viscous fluids, by assuming a point singularity in 3D. This
way of approaching the question has been taken very rarely over
the years. I will show, in the case of 3D Euler, that such a
point singularity is very unlikely because of the conservation
of energy and circulation, that puts very severe constraints
on the possible singularities. Following this line of thought,
one discovers that a singularity is possible along a line at
a given time. Furthermore, a pure self similar collapse is also
unlikely because it brings in a stationary flow with constant
positive divergence in the collapsing frame. If nothing counteracts
this positive divergence, anything is pulled to infinity in
the long run and there is no non trivial solution of the self
similar equations. Therefore one needs to have some sort of
instability that brings features at smaller and smaller scales
as one gets closer and closer to the singularity time. I'll
sketch a possible scenario for such a quasiself similar collapse
in axissymetric geometry.
Jacob
Rubinstein
(Department of Mathematics, Indiana University) jrubinst@indiana.edu
http://www.math.indiana.edu/people/profile.phtml?id=jrubinst
The
Least Action Principle for Waves
The
least action principle of Maupertuis, Lagrange, Hamilton and
others lies at the foundation of classical mechanics. Given
the initial and terminal points of a system of particles, the
orbit of the particles is determined by minimizing the action.
I shall describe a generalization of this principle that applies
to waves. The principle will be derived first for smooth solutions
of the Schroedinger equation, and then generalized to other
wave equations and to singular solutions.
Peter
Takac (Fachbereich
Mathematik, Universitaet Rostock ) takac@hades.math.unirostock.de
The
TimeDependent GinzburgLandau Equations in the Energy Space
W^{1,2} ()
(poster)
We
show the existence, uniqueness and wellposedness of mild solutions
to the timedependent GinzburgLandau equations (TDGL) in the
natural "energy space" W^{1,2} ().
This requirement is motivated by the fact that the GinzburgLandau
free energy should be finite. If the applied magnetic field
and its time derivative are bounded in time (e.g., timeperiodic)
as functions valued in L^{2}(),
then so is the solution of the TDGL equations. The solution
is strong for all positive times. To obtain our results, we
use abstract evolutionary equations with all terms valued in
L^{3/2} ().
The smoothing action of the heat semigroup is realized in L^{3/2}
().
Sylvia
Serfaty (Courant
Institute of Mathematical Sciences, New York University) serfaty@courant.nyu.edu
VortexDynamics
in the GinzburgLandau HeatFlow
The
GinzburgLandau equations are a model for superconductivity,
in which the crucial singularities are topological singularities
of vortex type. We will describe some results obtained with
Etienne Sandier on the dynamics of vortices in GinzburgLandau,
which are obtained by a Gammaconvergencebased method of convergence
which gives criteria to prove that solutions to the gradientflow
of energies which Gammaconverge to the solutions of the gradientflow
of the limiting energy. We will also describe some results on
vortexcollisions.
Jack
Xin (Department of Mathematics, University of Texas
 Austin) jxin@math.utexas.edu
The
(2+1) sineGordon Equation and Dynamics of Localized Pulses
Paper:
pdf
The (2+1) sineGordon (SG) equation is derived from a MaxwellBloch
system to model the dynamics of localized light pulses in two
space dimensional cubic nonlinear materials. A class of nonlinear
Schroedinger equations with both focusing and defocusing mechanisms
appears as underlying asymptotic description for both the propagation
and interaction phenomena.The dynamical persistence of pulses
is related to the internal oscillations. Alternative asymptotic
methods are reviewed to shed more light. Numerical simulations
show the robustness of solitary wave like interaction on the
plane.
Emmanuel
Yomba (Department of Physics, Faculty of Sciences,
University of Ngaoundere) eyomba@yahoo.com
New
exact Solutions for the Coupled KleinGordonSchoedinger Equations
(poster)
The
mapping method is used with the aid of symbolic computation
system Mathematica for constructing exact solutions to the coupled
KleinGordonSchroedinger equations. The solutions obtained
in this work include Jacobi elliptic solutions, combined Jacobi
elliptic solutions, triangular solutions, soliton solutions
and combined soliton solutions.
Wendy
Zhang
(Department of Mathematics and Statistics, Oakland University)
w2zhang@oakland.edu
http://www.oakland.edu/~w2zhang
Surface
Faceting Induced by Instability and Singularity in Evolution
Equations (poster)
We
study the effect of anisotropic surface free energy on the evolution
of surface morphology where the transport mechanism is surface
diffusion. Based on Herringâ^À^Ùs model, we formulate an evolution
equation, a partial differential equation, in a global coordinate
system. The partial differential equation is nonlinear, timedependent
and is of 4th order in space. The anisotropy is classified into
three cases, mild, critical and severe anisotropy, according
to the stability and singularity of the evolution equation.
We show that when the anisotropy is mild, the evolution equation
is stable and the surface is smooth. When the anisotropy is
critical, surface corners and edges appear in the evolution.
When the anisotropy is severe, small surface facets and coarsening
occur. The results of numerical computation of surface morphologies
are compared to experimental observations.
