Localized Multi-Dimensional Patterns in Dissipative Systems: Theory, Modeling, and Experiments (11w5091)

Arriving in Banff, Alberta Sunday, July 24 and departing Friday July 29, 2011


(University of Washington)

(Leiden University)

Edgar Knobloch (University of California Berkeley)

(Hokkaido University)

Bjorn Sandstede (Brown University)

(University of British Columbia)


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.

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.

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.

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.