1. Title: --------- Localized Multi-Dimensional Patterns in Dissipative Systems: Theory, Modeling, and Experiments. 2. ORGANIZERS: ------------- 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: ------------------ 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: -------- 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: ----------- 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: -------------- 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 ------------------- 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.