Applied Mathematical Sciences

Spectral and Dynamical Stability of Nonlinear Waves

Authors: Kapitula, Todd, Promislow, Keith

  • This book fills an important gap in the literature, bridging PDE and dynamical systems approach to stability
  • Presents a unified treatment of the dynamical systems and functional analysis background of nonlinear stability
  •  Includes illustrative examples and a variety of exercises
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Hardcover $79.95
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  • ISBN 978-1-4614-6994-0
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About this Textbook

This book unifies the dynamical systems and functional analysis approaches to the linear and nonlinear stability of waves. It synthesizes fundamental ideas of the past 20+ years of research, carefully balancing theory and application. The book isolates and methodically develops key ideas by working through illustrative examples that are subsequently synthesized into general principles.

Many of the seminal examples of stability theory, including orbital stability of the KdV solitary wave, and asymptotic stability of viscous shocks for scalar conservation laws, are treated in a textbook fashion for the first time. It presents spectral theory from a dynamical systems and functional analytic point of view, including essential and absolute spectra, and develops general nonlinear stability results for dissipative and Hamiltonian systems. The structure of the linear eigenvalue problem for Hamiltonian systems is carefully developed, including the Krein signature and related stability indices. The Evans function for the detection of point spectra is carefully developed through a series of frameworks of increasing complexity. Applications of the Evans function to the Orientation index, edge bifurcations, and large domain limits are developed through illustrative examples. The book is intended for first or second year graduate students in mathematics, or those with equivalent mathematical maturity. It is highly illustrated and there are many exercises scattered throughout the text that highlight and emphasize the key concepts. Upon completion of the book, the reader will be in an excellent position to understand and contribute to current research in nonlinear stability.

About the authors

Todd Kapitula is a Professor of Mathematics at Calvin College.  He previously held appointments at the University of New Mexico, Virginia Tech, the University of Utah, and Brown University.  He is the co-author of the 2008 Outstanding Paper Prize “Three is a crowd: solitary waves in photorefractive media with three potential wells”, SIAM J. Dyn. Sys. 5(4):598-633 (2006).  He is the author or co-author of over 40 research articles, and has been awarded several research grants from the National Science Foundation.

 

Keith Promislow is Professor of Mathematics at Michigan State University. His research interests include network morphology of amphiphilic systems induced  by charged-polymer solvent interactions. He serves on the editorial board of Physica D, SIAM Math Analysis, and SIAM Dynamical Systems. He represented the American Math Society at the Coalition for National Science Funding's 2011 Capital Hill Exhibit and was the 2010 Kloosterman Professor at the University of Leiden.

 

Reviews

From the book reviews:

“This 368-page book provides an excellent introduction to the spectral and nonlinear stability theory of nonlinear waves in one-dimensional domains. It is aimed primarily at graduate students, but it can certainly also be used by postdocs and other researchers who are interested in learning more about this area. … this book covers a broad range of topics in an area for which not many alternatives exist: the book is, in my opinion, an excellent addition to the literature.” (Bjorn Sandstede, Dynamical Systems Magazine, dynamicalsystems.org, April, 2014)

“The book under review focuses on stability of the equilibria of evolution equations. … The book consists of two parts. … Each chapter contains exercises, bibliographic comments and additional reading information. The reviewer evaluates the book as a significant achievement and recommends it to all interested readers.” (Radu Precup, zbMATH, Vol. 1297, 2014)

“This book by Kapitula and Promislow provides a quite unique entry point into this area, suitable for graduate students and young researchers who are interested in entering the field. … An extensive bibliography and plenty of remarks at chapter endings then serve as a guide to history and current literature. This field has needed such a book as an entry point for graduate students, and the authors deserve a huge thanks from the community for putting it together.” (Arnd Scheel, SIAM Review, Vol. 56 (3), 2014)


Table of contents (10 chapters)

  • Introduction

    Kapitula, Todd (et al.)

    Pages 1-4

  • Background Material and Notation

    Kapitula, Todd (et al.)

    Pages 5-37

  • Essential and Absolute Spectra

    Kapitula, Todd (et al.)

    Pages 39-74

  • Asymptotic Stability of Waves in Dissipative Systems

    Kapitula, Todd (et al.)

    Pages 75-115

  • Orbital Stability of Waves in Hamiltonian Systems

    Kapitula, Todd (et al.)

    Pages 117-157

Buy this book

eBook $59.99
price for USA (gross)
  • ISBN 978-1-4614-6995-7
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $79.95
price for USA
  • ISBN 978-1-4614-6994-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $79.95
price for USA
  • ISBN 978-1-4939-0187-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Spectral and Dynamical Stability of Nonlinear Waves
Authors
Series Title
Applied Mathematical Sciences
Series Volume
185
Copyright
2013
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4614-6995-7
DOI
10.1007/978-1-4614-6995-7
Hardcover ISBN
978-1-4614-6994-0
Softcover ISBN
978-1-4939-0187-6
Series ISSN
0066-5452
Edition Number
1
Number of Pages
XIII, 361
Topics