UK Nonlinear News Book Review
Pattern Formation: Introduction to Methods
Rebecca HoylePublished by Cambridge University Press (2006), 432 pp. ISBN-13: 9780521817509 | ISBN-10: 0521817501
Rebecca Hoyle's book is an invaluable graduate text for those who first enter the field of Pattern Formation. It has a remarkably lucid style and friendly exposition and very balanced level of detail through the book. An important strength is its rigorous mathematical approach which equips the reader with the underlying understanding of the mechanisms of pattern formation and enables one to confidently investigate novel problems. In this regard, I have been somewhat dissatisfied with the texts existing prior to this publication, at the time when I was starting my PhD course. The larger part of them had a distinctly physics flavour: presenting hard-to-get results but glossing over their mathematical justification. A few were concerned with the strictly theoretical development of the mathematical apparatus and were difficult to relate to application. A notable exception is maybe Golubitsky and Stewart's "The Symmetry Perspective", however I believe that Hoyle's book is more welcoming in terms of arrangement of topics and detail of presentation. An extensive part of the book is devoted to the problems of pattern formation in systems with non-compact symmetry groups which are not considered in the "The Symmetry Perspective".
I find it remarkable that a book presenting a field that has recently grown to such proportions, feels so light and approachable when you first take it in your hands. This is, I think, a much needed encouragement for the beginner. The uncluttered chapter structure and also the nice publisher design play a role in this. However it comes at a price - it might not be suitable to serve as a comprehensive reference bank for the field. Just as an example, of the results that I needed in my project two were not mentioned in the book, namely, the envelope equations for modulations with nonzero group velocity (mean-field Ginzburg-Landau equations), and the classification of chaos beyond Benjamin-Feir instability. Still, the book presents or refers to a surprising amount of the developments in theoretical Pattern Formation, and certainly, all the main concepts. Many novel results are described in essence and the reader is pointed to the relevant papers. A small number of the hardcore theoretical proofs are referred to the book of Golubitsky, Stewart and Schaeffer "Singularities and Groups in Bifurcation Theory". I think this improves on the readability of this volume.
The pace of the book is very well balanced, it gradually increases as the proficiency of the reader grows. Initially, in chapters 2 and 3, the necessary concepts of bifurcation analysis and group theory are nicely reviewed. As the equivariant bifurcation theory is introduced in chapter 4, the author works out in great detail the analysis of bifurcations in a square box. Heteroclinic cycles are also presented. Chapter 5 moves on to bifurcations on lattices, again, carefully describing the analysis for a square and a hexagonal lattice. By chapter 6 the methods for finding the symmetries, constructing and analysing the normal forms should be mastered and the author can concentrate on more complicated concepts - superlattice patterns, mode-interactions, quasipatterns and hidden symmetries. From chapter 7 to the end of the book the focus shifts to patterns unrestricted to a discrete lattice and the quite different approach of multiple scale expansion. The derivation of envelope equations from an underlying system as well as a general normal form describing a pattern with given symmetries (rolls and modulated hexagons) are described at length. Chapter 8 delves into the host of instabilities for rolls in 2D and 1D systems and chapter 9 describes instabilities of 2D patterns and quasipatterns. Chapter 10 looks at local defects such as roll dislocations and spirals. Finally, chapter 11 gives an overview of the Cross-Newell theory for variation of patterns in a large domain.
In conclusion, "Pattern Formation: Introduction to Methods" should be seen as an excellent textbook for the field, but not a review of all developments in it. It does contain a great number of techniques and methods illustrated through working out a number of problems important in the field.