UK Nonlinear News Review

Patterns and interfaces in dissipative dynamics

L. M. Pismen

Springer Series in Synergetics. Springer-Verlag, Berlin, 2006. xvi+369 pp. ISBN: 978-3-540-30430-2

Alastair Rucklidge

The 1980s and 90s were a period of intense development in the theory of pattern formation, and indeed the author of this book made important contributions. This book concentrates on three important classes of problems where significant progress was made during this time: the motion of fronts and interfaces, systems with two disparate time-scales (exemplified by the two-component reaction-diffusion problem, where one component diffuses much more rapidly than the other), and systems with two disparate spatial scales (such as steady or oscillatory patterns whose amplitudes very slowly in space). Within each category, a wide variety of possibilities is considered, making this book a comprehensive survey of the field.

Throughout, the material is presented from the point of view of the physicist, taking advantage of variational structure whenever this is present. Although not all problems are variational, at the level of a truncated amplitude equation, many problems do have variational structure, which can lead to considerable simplifications in the description of the dynamics. However, one consequence of this was that some interesting problems were not mentioned, and some interesting mathematical difficulties were hidden from view. For example, there was no mention of heteroclinic cycling, an interesting phenomenon that occurs in some pattern-forming systems and that is precluded by variational structure. There was little mention interesting non-adiabatic effects, where a long-range amplitude modulation can be pinned to the structure of an underlying pattern. Furthermore, little use is made of the comprenhensive understanding of the role of symmetry in pattern formation, as described in, for example, Golubitsky, Stewart and Schaeffer (Singularities and Groups in Bifurcation Theory, Volume II, 1988).

The book begins with a brief overview of dynamical systems theory, which is is helpful for understanding the bulk of the book. This introcuction includes the linear theory for reaction-diffusion systems, characterising the four important types of instability: whether this leads to steady or oscillatory growth, and whether the preferred wavenumber is zero or non-zero.

The second chapter deals with fronts between two patterns. In the simplest case these could be two metastable homogeneous stationary states. The stability of straight fronts is treated using the eikonal equation, which relates the speed of motion of the front to its curvature. This idea - the motion of a front treated as a geometric curve - recurs throughout the book. The case of front propagation into a region of unstable equilibrium is also treated.

Systems with disparate length scales are examined in chapter three. Examples include the Fitzhugh-Nagumo equation, as well as phenomenological models of fronts, which can describe the development of labyrinthine patterns. Spirals (and scroll waves) are treated as strongly coupled fronts.

The final two chapters examine the phase dynamics of stationary and oscillatory patterns. Patterns can be unstable to long-wave instabilities that alter the overall wavenumber by compressing or dilating the pattern (Ekhaus instability) or by twisting the pattern (zig-zag instability). Amplitude equations developed to treat these possibilties are premised on a separation between the scale of the pattern and the scale of the modulation. Notwithstanding this, the same amplitude equations are used to treat the evolution of defects, dislocations, grain boundaries and so on, an area where the author has made important contributions. Pinning effects are described, including the case of isolated hexagonal patterns. One important feature of oscillatory, travelling wave, patterns is that waves travelling in one direction at an order one speed, may have a slowly varying amplitude that is influenced by the averaged effects of waves travelling in the opposite direction.

In summary, this book is a clear and comprehensive review of this interesting area of pattern formation, and usefully complements existing texts. It is particularly useful as a reference to the development of the the theory of stability of fronts and defects.

This review first appeared in Mathematical Reviews. UK Nonlinear News is grateful to the American Mathematical Society for supplying a copy of the book.