Nonlinear Dynamics of Surface-Tension-Driven Instabilities Pierre Colinet, Jean Claude Legros, Manuel G. Velarde Wiley-VCH (2001), 527 pp., price USD 310.- (yes, that's not a typo) ISBN: 3527402918.
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.

Level: intermediate.

Surface tension effects and interfacial dynamics are important sources of nonlinear instability in numerous physical settings. Anyone who has gazed at the rainbow patterns of oil slicks in a parking lot puddle will appreciate the subtle beauty that a thin film can yield. Anyone who has tried to spray paint a metal surface evenly on a hot summer day will appreciate the dynamics that a thin layer of fluid can exhibit.

The starting point for this text is the well-known phenomenon of Rayleigh-Benard convection, in which the dynamics of a layer of fluid heated from below can bifurcate from dissipative heat transport to convective heat transport via `rolls' of alternating cells. The primary source of instability in the transition from conduction to convection is the Marangoni effect --- the change of surface tension with temperature. This effect allows tangential stresses to effect motion of the fluid at the surface.

From this starting point, the authors identify a plethora of fascinating phenomena with similar types of instabilities. For example, a thin layer of fluid can break up into convection cells with piecewise-linear interfaces ranging from square to hexagonal to more complex forms. Such systems are presumably at work in geological patterns seen in permafrost distribution and also in mineral distributions observable in evaporated lake beds. (The text contains some excellent pictures of this phenomenon.)

The body of the text is an analysis of instabilities in systems marked by such surface tension effects. Particular attention is paid to patterns and bifurcations.

The text is certainly suitable for a graduate-level topics course. Although the material is ostensibly orbiting about fluid-dynamical and thermo-dynamical phenomena, expertise in these topics is by no means prerequisite. The authors give an accessible introduction to the tools needed in the initial chapters of the text. Later chapters develop linear stability, and [weak] nonlinear instabilities, including monotonic and oscillatory instabilities. There is a strong experimental component to the text: this is one of its strengths.

There is also an effort to extend the ideas and applications to differential equations of a broader class. Later chapters give applications of the methods and techniques to real and complex Ginzburg-Landau equations, nonlinear Schroedinger equations, Kuramoto-Sivashinsky equations, and Korteweg-de Vries equations.

There is not much to complain about in this well-written, well-illustrated text, beyond the well-inflated list price. Anyone in applied dynamical systems will find this text written in a familiar language, which the Preface by I. Prigogine helpfully explains is ``the modern language of Nonlinear Physics.''