1. Title:
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Localized Multi-Dimensional Patterns in Dissipative Systems:
Theory, Modeling, and Experiments.
2. ORGANIZERS:
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Bernard Deconinck (Department of Applied Mathematics, U. Washington)
Arjen Doelman (Director of the Lorentz Center, Leiden U., The Netherlands)
Edgar Knobloch (Department of Physics, U. California, Berkeley)
Yasumasa Nishiura (Laboratory of Electronic Science, Hokkaido U.)
Bjorn Sandstede (Division of Applied Mathematics, Brown University)
Michael Ward (Department of Mathematics, UBC) (corresponding organizer)
3. TYPE OF MEETING:
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A five-day workshop at the Banff International Research Station is
proposed for either the week before or the week immediately after the
ICIAM 2011 conference in Vancouver.
4. SCOPE:
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This five-day workshop will provide a forum for the dissemination of
current advances in the mathematical analysis,
computational modeling, and experimental realizations of localized
patterns and coherent structures arising in fluids, nonlinear optics,
chemistry, and materials science. The aim is to bring together
theoreticians and experimentalists working on localized pattern
formation problems in diverse applications and from different
viewpoints to uncover common analytical or modeling approaches that
either advance our mathematical understanding or help
explain key experimental results relating to localized pattern
formation.
5. OVERVIEW:
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Spatially localized structures occur frequently in forced dissipative
systems. Well-studied examples include localized spots or cavity solitons
in a driven optical cavity, localized surface peaks in a ferrofluid
subjected to a normal magnetic field, and localized electrical breakdown
in a gas discharge. Other well-known examples include spatially localized
oscillations called oscillons, first identified in vibrating granular
media and subsequently observed in vibrating polymeric liquids. Convectons
or spatially localized convection are present in binary fluid mixtures
heated from below. Recent studies of the onset of shear flow turbulence
have revealed localized turbulence, in the form of stripes or patches,
prior to the development of the turbulent state. Buckling of slender
structures leads to spatially localized deformation. Other systems
such dewetting thin liquid films on a substrate or flowing over a
heated substrate exhibit some of the same behavior.
Experiments show that these structures may be stationary or move,
and have a tendency to form a variety of bound states resembling
molecules. In addition they may undergo instabilities leading to
fission, replication or disappearance. Such experiments often
show that distinct spatially localized structures coexist and
are simultaneously stable. This type of behavior generally occurs
in the region of bistability between a homogeneous or unstructured
background state and a spatially periodic or structured state: the
resulting localized states consist of an inclusion of the periodic
state embedded in the homogeneous background and connected to it by
fronts. These structures differ in their size, and numerical modeling
indicates that they lie on a small number of solution branches in a
bifurcation diagram that snake back and forth across part of the
bistability region. This mechanism for creating infinitely many
coexisting stable localized structures is commonly referred to as
snaking, and is now known to occur in the partial differential
equations of nonlinear optics and fluid dynamics. In one spatial
dimension it has been analyzed comprehensively via asymptotics beyond
all orders and dynamical-systems techniques in a fourth-order model
equation, the Swift-Hohenberg equation, which serves as a paradigm or
"normal form" for this phenomenon. However, many important questions
regarding snaking are still unresolved: for instance, except for a few
numerical studies, not much is known about the multi-dimensional case,
about the properties of bound states or indeed about time-dependent
structures such as oscillons.
Much recent effort has focused on reaction-diffusion systems, leading
to considerable advances in the theoretical understanding of the
dynamics and stability of localized pulses for two- and multi-component
reaction-diffusion systems in one spatial dimension. Equations such as
the Gray-Scott model are of interest in theoretical chemistry but related
equations arise in nonlinear optics, and electrical gas-discharge systems.
Theoretical tools to characterize the collision properties of pulses and
spots, and their interaction with spatial heterogeneities, have been developed
using computational global bifurcation approaches combined with dynamical
systems theory. Among the major challenges for the future are the
characterization of the stability and dynamics of localized structures
in reaction-diffusion systems on multi-dimensional domains.
Computational approaches such as numerical bifurcation and continuation
studies have played a significant role in discovering and illuminating
many of the phenomena mentioned above. Numerical computations have also
helped to understand experiments by facilitating comparisons with
models and by making predictions from models in cases where experiments
are difficult to carry out (examples include fluid flows and buckling).
There will be three main areas of focus during the workshop:
(I) The mathematical theory for the stability, dynamics, and
bifurcation properties of localized states in various PDE models,
including normal form systems such as the Swift-Hohenberg model.
(II) Numerical simulations, asymptotic theory, and mathematical models
characterizing localized patterns in various specific PDE systems
arising in applications.
(III) Real-world physical experiments and realizations of localized states
in fluid convection, optics, gas-discharge systems, and chemical systems.
The examination of such experimental phenomena by theoretical models.
Over the past 10 years there has been a growing interest in developing
new theoretical tools to analytically characterize the stability,
dynamics, and bifurcation properties of different types of localized
patterns in various PDE models, motivated by both numerical
simulations and physical experiments. There have been several
conferences in this direction including: The Newton Institute
Program from August--December 2005 on ``Pattern Formation in Large
Domains'', Cambridge University (organizers: J.H.P. Dawes, M. Golubitsky,
P. Matthews, A. Rucklidge); the week-long Japan-France international
conference ``Pattern Formation in Biology'' (organizers: K.-I.
Nakamura, M. Henry, M. Mimura) held at the University of Tokyo in
October 2005; the week-long international conference ``The Dynamics of
Patterns'' (organizers: A. Doelman, H. Broer) at the University of
Groningen, Holland, in April 2006; the Fields Institute workshop
``Patterns in Nonlinear PDE'' (organizers: W. Craig, C. Sulem,
N. Ercolani) held in Toronto in 2003; the BIRS workshop ``The Dynamics
of Localized Structures'' (organizers: P. Bates, T. Hillen, M. Ward,
J. Wei); the week-long workshop ``Patterns and Waves: Mathematics and
Nonlinear Chemistry'' (organizers: A. Doelman, Y. Nishiura), held at
the Lorentz Institute in Leiden in September 2001. Localized structures
also formed a core topic of the Oberwolfach Workshop ``Dynamics of Patterns''
held in December 2008 (organizers: W.-J. Beyn, B. Fiedler, B. Sandstede).
However, over the past several years a key emerging sub-area of
research focus in pattern formation has been the theoretical and
experimental characterization of localized pattern formation in
various diverse applications. To illustrate the importance and
relevance of this emerging research area, a three-day mini-course
``Multidimensional Localized Structures'' was given by four
distinguished lecturers (Ackemann (Strathcylde), Champneys (Bristol),
Knobloch (Berkeley), Scheel (Minnesota)) at the University of Rome in
2008. This mini-course was sponsored by the SIAM Activity Group on
Nonlinear Waves and Coherent Structures, and was held the weekend
before the international SIAM conference on Nonlinear Waves 2008, held
in Rome. One key theme in this conference was localized pattern
formation highlighted through a four part minisymposium ``Localized
Structures in Dissipative Systems I--IV'' (organizers: J.H.P. Dawes,
J. Burke), the minisymposium ``Self-Replication in Homogeneous Media''
(organizer: J. Rademacher); the minisymposium ``Pulse Dynamics in
Multi-Component Reaction-Diffusion Systems'' (organizers: A. Doelman,
T. Kaper); and the minisymposium ``Stability of Nonlinear Waves by
Computation'' (organizer: W. Beyn). Focused minisymposia on different
aspects of localized pattern formation were also hosted by the SIAM
Dynamical Systems Meeting at Snowbird in May 2009.
Despite this recent activity and interest in certain aspects of
localized pattern formation, to date there has not been a large-scale
international meeting for a broad-based examination and critical
overview of the recent advances made in both the theoretical
understanding and experimental realizations of localized patterns in
in a wide variety of contexts, such as nonlinear optics, fluid dynamics,
reaction-diffusion systems, normal form PDE models, granular media,
thin liquid film models of dewetting surfaces, etc... Our proposed
workshop will bring together leading international researchers in
various aspects of localized pattern formation, with a goal of
uncovering common theoretical approaches that can be used to characterize
localized states across a range of diverse applications.
6. OBJECTIVES:
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There are two main aims of this workshop. A primary goal is to enhance
the interaction between theoretical researchers in nonlinear aspects
of pattern formation with those researchers who are engaged in the
mathematical modeling or experimental realization of localized pattern
formation in diverse applications. The vast majority of these researchers
are regular participants in applied mathematics conferences and will be
attending the ICIAM Conference. This interaction should stimulate
new mathematical ideas, and also expose the mathematical community to
relevant new experimental situations involving localized patterns and
coherent states that await a theoretical understanding. A key feature
of this workshop is our intention to invite some noted experimentalists
who have observed and characterized localized patterns in
diverse real-world laboratory experiments. Our aim is to bring to the
forefront the sub-discipline of ``localized pattern formation'' as an
emerging and highly-active interdisciplinary area of pattern formation
with many challenging and interesting open directions.
The second main focus of the workshop is to expose a limited number of
Postdoctoral Fellows and advanced graduate students to current
problems associated with localized pattern formation, and to highlight
some of the recent mathematical advances in stability and bifurcation
theory, dynamical systems, asymptotic analysis, and PDE theory used to
study this behavior.
The proposed ICIAM satellite training workshop titled ``Stability and
Instability of Coherent Structures and Patterns'' (organizers:
B. Deconinck, S. Gustafson, and M. Ward), submitted to ICIAM and to be
held before our proposed BIRS workshop, will provide some of the
necessary background mathematical material on stability theory for our
more junior BIRS workshop participants.
7. SPECIFIC FORMAT OF THE WORKSHOP
----------------------------------
The expertise of each of the participants listed below is well-matched
to one of the three highlighted core areas of the workshop, with
roughly equal representation in these three areas. For each core area
we will invite a keynote speaker from the participant list to give a
two-hour survey lecture. These lectures should provide an overview of
important topics and advances in each of the core areas, and will
highlight open problems of either a mathematical, modeling, or
experimental focus, that await investigation. The survey talks in
each of these three different areas of focus will provide a key forum
for facilitating a lively scientific exchange between the relatively
diverse group of participants.
It is intended that the keynote speakers will prepare a written survey
of their lectures for dissemination, and be videotaped by BIRS for
wider off-site distribution. Each of these three keynote lectures will
be followed by a series of 45-minute lectures dealing with various
specific problems in the field. On the final afternoon of the workshop
a round-table discussion will identify possible future
collaborations and links, and suggest new areas to focus future
research directions in the theoretical understanding of localized
patterns. As a way of unifying and collecting some of the rather
diverse material on the more mathematical aspects relating to localized
pattern formation, the organizers will attempt to publish a dedicated
special issue on this topic in a leading international applied
mathematics journal. One candidate for such a special issue is the
``The European Journal of Applied Mathematics'', published by Cambridge
U. Press, and affiliated with Oxford University, which has expressed a
keen interest in this project.
8. Participant List
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T Ackemann (Strathclyde)
N Akhmediev (Canberra)
M Beck (Boston)
A Belmonte (Penn State)
A Bergeon (Toulouse)
W Beyn (Bielefeld)
J Burke (Boston)
A Champneys (Bristol)
SJ Chapman (Oxford)
M Clerc (Santiago de Chile)
A Cliffe (Nottingham)
JHP Dawes (Bath)
B Deconinck (U. Washington)
A Doelman (Leiden)
B Eckhardt (Marburg)
S Ei (Kyushu)
I Epstein (Brandeis)
W Firth (Strathclyde)
A Ghazaryan (Kansas)
K Glasner (Arizona)
D Gomila (Palma de Mallorca)
M Haragus (Besancon)
S Houghton (Leeds)
M Iima (Hokudai)
D Iron (Dalhousie)
CKRT Jones (UNC/Warwick)
T Kaper (Boston)
E Knobloch (Berkeley)
J Knobloch (Ilmenau)
T Kolokolnikov (Dalhousie)
G Kozyreff (Brussels)
N Kutz (U. Washington)
D Lloyd (Surrey)
H Mahara
I Mercader (Barcelona)
E Meron (Negev U.)
A Meseguer (Barcelona)
M Mimura (Meiji)
T Mullin (Manchester)
Y Nishiura (Hokkaido)
K Nishi (Hokudai)
T Ogawa (Osaka)
M Peletier (Eindhoven)
K Promislow (Michigan State)
HG Purwins (Muenster)
J Rademacher (CWI, Amsterdam)
X Ren (George Washington U.)
S Residori (Nice)
R Richter (Bayreuth)
B Sandstede (Brown)
A Scheel (Minnesota)
G Schneider (Stuttgart)
T Schneider (Harvard)
H Swinney (U. Texas, Austin)
S Tavener (Colorado State)
T Teramoto (Chitose)
U Thiele (Loughborough)
L Tuckerman (PMMH, Paris)
K Ueda (Kyoto)
D Ueyama (Meiji)
T Yamaguchi
P Van Heijster (Brown)
T Wagenknecht (Leeds)
J Wei (Chinese U. Hong Kong)
The participant list includes both senior researchers and junior
faculty who have made key contributions to either the theoretical
understanding, computational modeling, or experimental realizations,
of localized pattern formation. We have contacted roughly 20
established faculty members who have expressed their keen interest to
participate and contribute to the workshop should it be funded. We
expect that approximately 30--35 well-established researchers will be able
to attend the workshop. This will allow approximately 5--10 slots for
postdoctoral fellows and advanced graduate students. The total number
of participants at the workshop will be 40.