This Research Note aims to provide an insight into recent developments in the theory of pattern formation. In the last decade there has been considerable progress in this field, both from a theoretical and a practical point of view. Recent mathematical developments concern the study of the nonlinear stability of systems at near-critical conditions by an appropriate system of modulation equations. The complexity of the original problem can be reduced drastically by this approximation. Moreover, it provides unifying point of view for a wide range of problems. New applications of the theory arise in a multitude of scientific areas such as hydrodynamics, reaction-diffusion problems, oceanography, meteorology, combustion, geophysical and biological morphodynamics and semi-conductors.
This book is intended to show the interactions between the mathematical theory of nonlinear dynamics and the study of pattern generating phenomena in the natural environment. There is an intimate relationship between new insights in the mathematical aspects of nonlinear pattern formation and the comprehension of such phenomena. Therefore there are two partly overlapping main themes: one in which the emphasis is on generally applicable mathematical theories and techniques and one in which the phenomenology of pattern evolution in various areas is discussed.
The book comprises 19 contributions by experts in the field. Although the emphasis changes considerably from paper to paper, in each contribution the same two themes are present; all the authors have aimed to achieve a suitable balance between the mathematical theory and the physical phenomena.
Table of Contents
Nonlinear evolution of cellular flame instabilities
On the justification of the Ginzburg-Landau approximation
Bifurcations far from criticality in fluid systems
Self-organization and chaotic advection in quasi-2D confined
Surface tension driven cellular flows in small aspect ratio horizontally square boxes
Pattern formation in systems on spatially periodic domains
Solitary wave interactions with external forces
Breaking the dimension of a steady wave: some examples
Remarks on the use and misuse of the GinzburgñLandau equation
The mean flows driven by sandbar instabilities
Dynamical systems, temporal vs. spatio-temporal chaos, and climate
Interaction of modes with disparate scales in Rayleigh-Benard convection
A new approach to sideband-instabilities using the principle of reduced instability
Pattern formation in activator-inhibitor systems
Instabilities in two-layer channel flows
A personal sample of patterns in biology
Invitation to river morphodynamics
Periodic orbits in singularly-perturbed systems
Dynamics of large-scale bed forms in coastal seas
"An impressive selection of articles by leading experts in the field....researchers and more advanced students in mathematical analysis will find it a very valuable addition to the existing literature and a rich source of challenges for the further development of mathematical theory"
Nonlinear Science Today, September 1996