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Traveling Wave Solutions of Parabolic Systems
Aizik I. Volpert, Technion-Israel Institute of Technology, Haifa, Israel, Vitaly A. Volpert, University of Lyon I, Villeurbanne Cedex, France, and Vladimir A. Volpert, Northwestern University, Evanston, IL
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Translations of Mathematical Monographs
1994; 448 pp; softcover
Volume: 140
Reprint/Revision History:
reprinted 2000
ISBN-10: 0-8218-1143-6
ISBN-13: 978-0-8218-1143-6
List Price: US$132 Member Price: US$105.60
Order Code: MMONO/140.S

The theory of traveling waves described by parabolic equations and systems is a rapidly developing branch of modern mathematics. This book presents a general picture of current results about wave solutions of parabolic systems, their existence, stability, and bifurcations. The main part of the book contains original approaches developed by the authors. Among these are a description of the long-term behavior of the solutions by systems of waves; construction of rotations of vector fields for noncompact operators describing wave solutions; a proof of the existence of waves by the Leray-Schauder method; local, global, and nonlinear stability analyses for some classes of systems; and a determination of the wave velocity by the minimax method and the method of successive approximations. The authors show that wide classes of reaction-diffusion systems can be reduced to so-called monotone and locally monotone systems. This fundamental result allows them to apply the theory to combustion and chemical kinetics. With introductory material accessible to nonmathematicians and a nearly complete bibliography of about 500 references, this book is an excellent resource on the subject.

Mathematicians studying systems of partial differential equations, reaction-diffusion systems; physicists interested in autowave processes, dissipative structures; combustion scientists and chemists interested in mathematical issues of chemical kinetics.

Reviews

"A well-written and welcome addition to the literature on this subject ... much of the text presents the results of papers that are only available in Russian and are not easily accessible to Western readers. Most of the book would be suitable for graduate students after a course in P.D.E., and the first part of the book contains an excellent introduction to the elementary aspects of the subject."

-- Bulletin of the AMS

"A well-written book which covers in a very satisfying way a difficult subject matter, providing ... a well-balanced interplay between theory and applications ..."

-- Mathematical Reviews

Part I. Stationary waves
• Scalar equation
• Leray-Schauder degree
• Existence of waves
• Structure of the spectrum
• Stability and approach to a wave
Part II. Bifurcation of waves
• Bifurcation of nonstationary modes of wave propagation
• Mathematical proofs
Part III. Waves in chemical kinetics and combustion
• Waves in chemical kinetics
• Combustion waves with complex kinetics
• Estimates and asymptotics of the speed of combustion waves
• Asymptotic and approximate analytical methods in combustion problems (supplement)
• Bibliography