(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
This book is based on an undergraduate course taught at the IAS/Park City Mathematics Institute, on linear and nonlinear waves. The first part of the text overviews the concept of a wave, describes one-dimensional waves using functions of two variables, provides an introduction to partial differential equations, and discusses computer-aided visualization techniques.
The second part of the book discusses traveling waves, leading to a description of solitary waves and soliton solutions of the Klein-Gordon and Korteweg-deVries equations. The wave equation is derived to model the small vibrations of a taut string, and solutions are constructed via d'Alembert's formula and Fourier series.
The last part of the book discusses waves arising from conservation laws. After deriving and discussing the scalar conservation law, its solution is described using the method of characteristics, leading to the formation of shock and rarefaction waves. Applications of these concepts are then given for models of traffic flow.
The intent of this book is to create a text suitable for independent study by undergraduate students in mathematics, engineering, and science. The content of the book is meant to be self-contained, requiring no special reference material. Access to computer software such as Mathematica, MATLAB, or Maple is recommended, but not necessary. Scripts for MATLAB applications will be available via a Web site. Exercises are given within the text to allow further practice with selected topics.
Student Mathematical Library
October 1999 (estimated), approximately 200 pages, Softcover,
This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrödinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with large Cauchy data. Current research in harmonic analysis around Strichartz's inequalities and its relevance to nonlinear PDE is presented. Several topics in NLS theory on bounded domains are reviewed. Using the NLS as an example, the book offers comprehensive insight on current research related to dispersive equations and Hamiltonian PDEs.
Colloquium Publications, Volume 40
July 1999, 182 pages, Hardcover
This book presents developments in the geometric approach to nonlinear partial differential equations (PDEs). The expositions discuss the main features of the approach, and the theory of symmetries and the conservation laws based on it. The book combines rigorous mathematics with concrete examples. Nontraditional topics, such as the theory of nonlocal symmetries and cohomological theory of conservation laws, are also included.
The volume is largely self-contained and includes detailed motivations, extensive examples and exercises, and careful proofs of all results. Readers interested in learning the basics of applications of symmetry methods to differential equations of mathematical physics will find the text useful. Experts will also find it useful as it gathers many results previously only available in journals.
Translations of Mathematical Monographs,
Volume 182; 1999
approximately 347 pages; Hardcover; ISBN 0-8218-0958-X
A corrected paperback version of this book is now available in paperback (ISBN 0-521-65387-8, price #19.95). It was reviewed by Jaroslav Stark in UK Nonlinear News 13 (August 1998).
Over the last two decades, chaos in engineering systems has moved from being simply a curious phenomenon to one with real, practical significance and utility. Engineers, scientists, and mathematicians have similarly advanced from the passive role of analyzing chaos to their present, active role of controlling chaos - control directed not only at suppression, but also at exploiting its enormous potential. We now stand at the threshold of major advances in the control and synchronization of chaos for new applications across the range of engineering disciplines.
Controlling Chaos and Bifurcations in Engineering Systems provides a state-of-the-art survey of the control and anti-control of chaos in dynamical systems. Internationally known experts in the field join forces in this volume to form this tutorial-style combination of overview and technical report on the latest advances in the theory and applications of chaos control. They detail various approaches to control and show how designers can use chaos to create a wider variety of properties and greater flexibility in the design process.
Chaos control promises to have a major impact on novel time- and energy-critical engineering applications. Within this volume, readers will find many challenging problems - yet unsolved - regarding both the fundamental theory and potential applications of chaos control and anti-control. Controlling Chaos and Bifurcations in Engineering Systems will bring readers up-to-date on recent development in the field and help open the door to new advances.
Audience: Electrical Engineers, Mechanical Engineers, Systems and Controls Engineers, Applied Mathematicians, Applied Physicists.
On the second edition "The subject has wide applications in physical, biological, and social sciences which continuously supply new problems of practical and theoretical importance. The book does a good job in motivating the reader in such pursuits, and presents the subject in a simple but elegant style." - P.K. Kythe in Applied Mechanics Reviews. A standard text book in the field, this volume takes a qualitative approach and includes numerous examples and problems. The new edition features expanded material on bifurcation and chaos.
September 1999. 416pp. 230 line illustrations.
Resonances are ubiquitous in dynamical systems with many degrees of freedom. They have the basic effect of introducing slow-fast behaviour in an evolutionary system which, coupled with instabilities, can result in highly irregular behaviour. This book gives a unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, a general finite dimensional theory of homoclinic jumping is developed and illustrated with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context. Previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds are described. The results are applied to a variety of different problems, which include applications from beam oscillations, surface wave dynamics, nonlinear optics, atmospheric science and fluid mechanics. The theory is further used to study resonances in Hamiltonian systems with applications to molecular dynamics and rigid body motion. The final chapter contains an infinite dimensional extension of the finite dimensional theory, with application to the perturbed nonlinear Schrödinger equation and coupled NLS equations.
Contents: Concepts from dynamical systems.- Chaotic jumping along resonances: Finite dimensional systems.- Homoclinic Chaos due to Resonances in Physical Systems.- Resonances in Hamiltonian systems.- Chaotic jumping along resonances: Evolution equations.- A Elements of differential geometry.- B Some facts from analysis.
Applied Mathematical Sciences, Volume 138
1000. Approx 455pp. 155 figures.
Time-evolution in low-dimensional topological spaces is a a subject of puzzling vitality. This book is a state-of-the-art account, covering classical and new results. The volume comprises Poincaré-Bendixson, local and Morse-Smale theories, as well as a carefully written chapter on the invariants of surface flows. Of particular interest are chapters on the Anosov-Weil problem, C*-algebras and non-compact surfaces. The book invites graduate students and non-specialists to a fascinating realm of research. It is a valuable source of reference to the specialists.
Keywords: Closed Surface Flow, Ergodic Measure, Irrational Rotation Algebra.
Lectures Notes in Mathematics Volume 170
1999, 294pp, 54 figures
The purpose of this book is to promote the transfer of information between the various communities concerned with nonlinear waves. Special attention is paid to the phenomenon of self-focusing and wave collapse. Various approaches ranging from rigorous mathematical analysis to formal asymptotic expansions and numerical simulations are presented. An extended and up-to-date bibliography is included. Graduate students and researchers in the fields of pure and applied mathematics, nonlinear optics, plasma physics, hydrodynamics, and magnetohydrodynamics will find this book useful.
Applied Mathematical Sciences, Volume 139
1999/368PP., 9 illustrations.
This volume contains a number of mini-review articles authored by speakers and attendees at the IMA workshop on Pattern Formation in Continuous and Coupled Systems. The reviews in this volume are intended to be pointers to the current literature for researchers at all levels and therefore include extensive bibliographies.
The IMA Volumes in Mathematics and its Applications Volume 115
1999/App 344PP, 101 illustrations.
This book is intended for an audience of researchers and graduate students from stochastics as well as dynamics. It gives an account of new developments in the theory of random and stochastic dynamical systems. Recent results in stochastic bifurcation, hyperbolic systems, numerics and asymptotics, more general driving processes for stochastic differential equations, and stochastic analysis on infinite-dimensional manifolds are presented in a comprehensible manner. Also, several new and exciting insights into the unexpected variety of dynamical behaviours resulting from the influence of stochastic perturbations are conveyed to the reader.
1999. 472pp. 29 illustrations