UK Nonlinear News, August 2001
World Scientific Press
300 pages, hardcover.
This book contains a number of survey papers on topics in nonlinear mathematical physics and mathematical biology, based on lectures delivered during two INLN (Institut Nonlinear de Nice) Summer Schools on Nonlinear Phenomena, held June 1998 and June 1999 in Peresq, a small village in the French Alps.
By compiling and publishing lecture notes out of the summer schools, the editors Kaiser and Montaldi have created an audience for the material presented, that reaches far beyond the (small) one present during the schools themselves. Although a proven concept, the succesful compilation of such lecture notes is not a simple task. It requires not only the selection of topics with appealing content (for the school), but also enthousiasm and endurance to convince lectureres to contribute good quality lecture notes. I feel that the book under review has succeeded quite well delivering on those aims. As is to be expected, some papers dealt with topics I knew more of then others, but to me, all came across as interesting and contained extensive bibliographies for further reading.
The contributed papers deal with various aspects of the theoretical study of nonlinear phenomena, during the stages of modelling and the mathematical analysis of models.
The paper 'Elasticity and Geometry' by Audoly and Pomeau deals with the very fundamentals of deriving F\"oppl-von Karman equations for thin plate elasticity, focussing on differential geometric aspects, and some special solutions of these equations. Their discussion is very elementary and was a pleasure to read (for myself as a non-expert).
D. Delande's contribution is an elementary introduction to the field of 'Quantum Chaos'. This field deals with phenomena in quantum mechanics that have parallels with chaotic behaviour in classical mechanical systems. The paper discusses how semi-classical approximations to quantum mechanical systems may lead to an understanding of quantum chaos. In particular, the random matrix theory for the statistical properties of energy levels of 'chaotic' quantum systems is discussed, as well as the '(Gutzwiller) Trace Formula' describing a relation between periodic orbits of a classical mechanical system and the density of states of a related 'chaotic' quantum mechanical system. The discussion is elementary and easy to read for the non-expert with a basic knowledge of classical and quantum mechanics.
The practical relevance of 'quantum chaos' has - to my knowledge - yet to be demonstrated. Robin Kaiser's contribution 'Cold Atoms and Multiple Scattering' demonstrates that there are still many more elementary phenomena to be understood of quantum systems in the semi-classical regime. The paper describes a (semi-)classical model for laser-cooling of atoms. It is well written, though more interesting for those of us who have a basic knowledge of atomic (quantum) physics.
In 'The Water-wave Problem as a Spatial Dynamical System', G. Iooss discusses how the water-wave problem has led to (years of intensive) studies of bifurcations in reversible dynamical systems. The water-wave problem deals with the description of the spatial behaviour of travelling waves near a basic flat free surface state of water (modelling it as an inviscid fluid). Assuming finite depth, the model is obtained by reduction of the relevant PDE to an ODE by assuming travelling waves with constant speed and a center-manifold reduction. Reversibility of the latter ODE arises as a consequence of a spatial reflection symmetry assumed in the water-wave problem. The problem is then studied using normal form analysis. The latter (formal) analysis poses some problems, as it turns out the important issue of the existence of homoclinic solutions near a bifurcation point cannot always be decided from the (formal) normal form analysis, as the fine details may be hidden in the ignored (flat) terms of arbitrarily high order.
The latter problem is tackled by Lombardi, whose contribution 'Phenomena Beyond All Order and Bifurcations of Reversible Homoclinic Connections near Higher Resonances' deals precisely with the study of homoclinic solutions whose existence cannot be decided by a formal normal form analysis. The heart of Lombardi's toolbox is complex analysis. Interestingly, his study has shown that solatory solutions (arising as homoclinic solutions in normal forms of reversible systems) often turn out to be unstable to flat perturbations (that are ignored in formal normal forms). The problem arises as well in many other problems of physical interest. For an in-depth discussion, the reader is advised to consult Lombardi's monograph (volume 1741 of Lecture Notes in Mathematics, Springer, 2000).
The contribution of J. Montaldi, 'Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems', provides an introduction to the study of relative equilibria in symmetric Hamiltonian (classical mechanical) systems. Relative equilibria are solutions entirely contained inside orbits of the symmetry group of the system (and equilibria in the reduced phase space). The paper provides a very readable introduction into the basic mathematical tools for studying existence and local bifurcation of relative equilibria.
Finally, in 'Mathematical Modelling in the Life Sciences: Applications in Pattern Formation and Wound Healing', P. Maini surveys the topical study of mathematical models (reaction-diffusion equations) for pattern formation in biology and wound-healing, introducing various models and their applications.
A listing of books reviewed in UK Nonlinear News is available.
UK Nonlinear News thanks World Scientific for providing a review copy of this book.