(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
The study of asymptotic solutions to nonlinear systems of partial differential equations is a very powerful tool in the analysis of such systems and their applications in physics, mechanics, and engineering. In the present book, the authors propose a new powerful method of asymptotic analysis of solutions, which can be successfully applied in the case of the so-called "smoothed shock waves", i.e., nonlinear waves which vary fast in a neighbourhood of the front and slowly outside of this neighbourhood. The proposed method, based on the study of geometric objects associated to the front, can be viewed as a generalization of the geometric optics (or WKB) method for linear equations. This volume offers to a broad audience a simple and accessible presentation of this new method.
The authors present many examples originating from problems of hydrodynamics, nonlinear optics, plasma physics, mechanics of continuum, and theory of phase transitions (free boundary problems). In the examples, characterized by smoothing of singularities due to dispersion or diffusion, asymptotic solutions in the form of distorted solitons, kinks, breathers, or smoothed shock waves are constructed. By a unified rule, a geometric picture is associated with each physical problem that allows for obtaining tractable asymptotic formulas and provides a geometric interpretation of the physical process. Included are many figures illustrating the various physical effects.
Translations of Mathematical Monographs, Volume 202
August 2001, 285 pages
Hardcover, ISBN 0-8218-2109-1
During the past decade, there have been several major new developments in smooth ergodic theory, which have attracted substantial interest to the field from mathematicians as well as scientists using dynamics in their work. In spite of the impressive literature, it has been extremely difficult for a student--or even an established mathematician who is not an expert in the area--to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools.
Accordingly, the AMS Summer Research Institute on Smooth Ergodic Theory and Its Applications (Seattle, WA) had a strong educational component, including ten mini-courses on various aspects of the topic that were presented by leading experts in the field. This volume presents the proceedings of that conference.
Smooth ergodic theory studies the statistical properties of differentiable dynamical systems, whose origin traces back to the seminal works of Poincaré and later, many great mathematicians who made contributions to the development of the theory. The main topic of this volume, smooth ergodic theory, especially the theory of nonuniformly hyperbolic systems, provides the principle paradigm for the rigorous study of complicated or chaotic behaviour in deterministic systems. This paradigm asserts that if a non-linear dynamical system exhibits sufficiently pronounced exponential behaviour, then global properties of the system can be deduced from studying the linearized system. One can then obtain detailed information on topological properties (such as the growth of periodic orbits, topological entropy, and dimension of invariant sets including attractors), as well as statistical properties (such as the existence of invariant measures, asymptotic behaviour of typical orbits, ergodicity, mixing, decay of correlations, and measure-theoretic entropy). Smooth ergodic theory also provides a foundation for numerous applications throughout mathematics (e.g., Riemannian geometry, number theory, Lie groups, and partial differential equations), as well as other sciences.
This volume serves a two-fold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a state-of-the-art report on important current aspects of the subject. The book is divided into three parts: lecture notes consisting of three long expositions with proofs aimed to serve as a comprehensive and self-contained introduction to a particular area of smooth ergodic theory; thematic sections based on mini-courses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.
October 2001, 867 pages,
hardcover, ISBN 0-8218-2682-4
This work, consisting of expository articles as well as research papers, highlights recent developments in nonlinear analysis and differential equations. Several topics in ordinary differential equations and partial differential equations are the focus of key articles, including:
A number of related subjects dealing with properties of solutions, e.g., bifurcations, symmetries, nonlinear oscillations, are treated in other articles.
This volume reflects rich and varied fields of research and will be a useful resource for mathematicians and graduate students in the ODE and PDE community.
Progress in Nonlinear Differential Equations and their Applications, Volume 43
2001, 456 pages
ISBN 0-8176-4188-2, Hardcover
Nonlinear physics continues to be an area of dynamic modern research, with applications to physics, engineering, chemistry, mathematics, computer science, biology, medicine and economics. In this text extensive use is made of the Mathematica computer algebra system. No prior knowledge of Mathematica or programming is assumed. The authors have included a CD-ROM that contains over 130 annotated Mathematica files. These files may be used to solve and explore the text's 400 problems. This book includes 33 experimental activities that are designed to deepen and broaden the reader's understanding of nonlinear physics. These activities are correlated with Part I, the theoretical framework of the text.
Additional features:
This work is an excellent text for undergraduate and graduate students as well as a useful resource for working scientists.
2001, 704 pages, 282 illustrations
ISBN 0-8176-4223-4, Hardcover with CD-ROM
Synchronization phenomena are abundant in science, nature, engineering and social life. Systems as diverse as clocks, singing crickets, cardiac pacemakers and applauding audiences exhibit a tendency to operate in synchrony. This universal phenomenon can be understood within a common framework based on modern nonlinear dynamics. The book describes synchronization first without formulae, using experimental examples, and then in a rigorous and systematic manner, describing both classical results and recent developments. This comprehensive text will be of interest to graduate students and specialist researchers.
For more information visit:
http://www.cambridge.org/physics/catalogue/0521592852
A special mathematics calender called MathInsight 2002 has been published. It offers spectacular visual presentations of mathematics. Mathematicians from different fields and experts together went to the limit to give you the ideal add-on for your office.
2001. Calendar with 14 pp. 14 figs, in colour, A2 size
ISBN 3-540-67472-1
DM 49,90 (net price subject to local VAT)
http://www.springer.de/cgi-bin/bag_generate.pl?ISBN=3-540-67472-1