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© 2013

Spectral and Dynamical Stability of Nonlinear Waves

  • 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

Textbook

Part of the Applied Mathematical Sciences book series (AMS, volume 185)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Todd Kapitula, Keith Promislow
    Pages 1-4
  3. Todd Kapitula, Keith Promislow
    Pages 5-37
  4. Todd Kapitula, Keith Promislow
    Pages 39-74
  5. Todd Kapitula, Keith Promislow
    Pages 75-115
  6. Todd Kapitula, Keith Promislow
    Pages 117-157
  7. Todd Kapitula, Keith Promislow
    Pages 159-175
  8. Todd Kapitula, Keith Promislow
    Pages 177-213
  9. Todd Kapitula, Keith Promislow
    Pages 215-247
  10. Todd Kapitula, Keith Promislow
    Pages 249-304
  11. Todd Kapitula, Keith Promislow
    Pages 305-344
  12. Back Matter
    Pages 345-361

About this book

Introduction

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.

Keywords

Evans function Hamiltonian systems Lyapunov-Schmidt reductions Nonlinear Waves Spectral Theory

Authors and affiliations

  1. 1., Department of Mathematics and StatisticsCalvin CollegeGrand RapidsUSA
  2. 2., Department of MathematicsMichigan State UniversityEast LansingUSA

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.

 

Bibliographic information

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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)