`UK Nonlinear News`,
`November 2001`

Lecture Notes in Physics series.

Springer, Berlin, 1998, 583 pp.

hardback, ISBN 3-540-65154-3.

Pattern formation occurs in many systems in nature. Classic examples involve fluid instabilities driven by buoyancy, vibration or shear, but there are many other involving, for instance, electric or magnetic fields, chemical reactions, biological systems and even granular materials. While the physics of each problem may be different, there is enough mathematical similarity that a general nonlinear theory of pattern formation, or spontaneous formation of structure, has developed.

This edited volume aims to demonstrate, by providing numerous examples, that the common mathematical language of pattern formation can fruitfully be used to interpret the transition from order to disorder in a wide variety of physical systems. The eponymous book originated as part of a six-year program on the `Evolution of Spontaneous Structures in Dissipative Continuous Systems', funded by the Deutsche Forschungsgemeinschaft, and the intention was to review the current state of the field using the projects in the research program as examples. Many, but not all, of the contributors to the volume were part of the program, but not all projects in the program were included in the volume.

The book is made up of two main parts. The first fifth or so is a review article `Mathematical tools for pattern formation' by Dangelmayer and Kramer, providing the background mathematical structure for the remainder of the volume. There are two main streams within the mathematical theory, and which one is relevant depends crucially on the symmetry of the physical system under consideration. Closed experimental systems, or numerical calculations posed in periodic domains, will result in finite sets of ordinary differential equations (ODEs), the range of validity of which decreases as the size of the system increases. Open-flow experiments, or calculations performed in large (strictly speaking, infinite) domains, result in sets of partial differential equations (PDEs), typically of the Ginzburg-Landau type. This dichotomy is reflected in the structure of the first review article: the first half discusses pattern-forming transitions in systems that have discrete or circular symmetries, and the second discusses the real and complex Ginzburg--Landau equation in one and two dimensions, and related equations.

The second, and much larger, part of the book touches on many important areas of pattern formation in fluid mechanical and related problems. Each chapter describes the recent work of the chapter authors and, to a greater or lesser extent, reviews the work of other researchers. The topics covered include: the Taylor-Couette system and others characterized by an axisymmetric experimental setup; binary fluid convection; pattern formation with through-flow; coherent structure formation in open flows; theoretical and experimental aspects of pattern formation in the Faraday wave experiment in both small and large containers; pattern formation in the presence of inhomogeneities in the background state; pattern formation in electrically driven smectic and nematic liquid crystals; spiral wave formation during surface catalysed and auto-catalytic chemical reactions; pattern formation in charge carriers in semi-conductors; pattern formation in granular materials; generation of magnetic fields in a laboratory dynamo; and the self-organised patterns of the slime-mold Dictyostelium discoideum. Many of the articles combine theoretical, numerical and experimental aspects and have lengthy lists of references.

Overall, the book is a remarkable achievement, bringing together as it does so many of the strands of research in so many different fields of physics and mathematics, and managing to present a digestible mixture of experiment and theory in almost every article. The quality of the individual articles is good overall but varies: some have been written as general reviews of the field, and so should be useful to a wide audience, while others are more restricted in scope and will have a shorter shelf-life. The review of developments in the Faraday wave experiment, by Muller, Friedrich and Papathanassiou, is particularly useful to me, but other readers will find interest in other places. I also thought it interesting to see the relative position of theory and experiment in different fields: in the Faraday problem, for instance, quantitative comparisons between the nonlinear theory and the observed experimental pattern are possible, while in electroconvection, even the linear theory is too difficult owing to the large number of material parameters.

The part of the book I find the least satisfying is the introductory review of the mathematical background: the scope is broad, but not enough space has been given to the review to do the topics justice, and many of the technical ideas need more explanation to make them intelligible. But I also think that an opportunity has been missed. As mentioned above, the first half of the review deals with finite-dimensional dynamics (ODEs) and the second with infinite-dimensional dynamics (PDEs) -- there is much current interest in the boundary between these two, relevant to pattern formation in large but finite boxes. This would have been an ideal place for an exploration of topics such as the formation of quasi-patterns, which have orientational but not translational order.

The back cover states that the book `addresses researchers but it could also be used as a text for graduate work'. I agree with the first part of this but not the second, though of course individual articles could be useful to those starting out in this field.

This review was commissioned by the `Journal of Fluid Mechanics`, and it
should appear there shortly. It is reprinted in `UK Nonlinear News`
with the permission of the `JFM` book review editor.

A listing of books reviewed in `UK Nonlinear News`
is available.

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Last Updated: 10th October 2001.