Spectral and Dynamical Stability of Nonlinear Waves

  • Todd Kapitula
  • Keith Promislow

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

  • Todd Kapitula
    • 1
  • Keith Promislow
    • 2
  1. 1., Department of Mathematics and StatisticsCalvin CollegeGrand RapidsUSA
  2. 2., Department of MathematicsMichigan State UniversityEast LansingUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-6995-7
  • Copyright Information Springer Science+Business Media New York 2013
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-6994-0
  • Online ISBN 978-1-4614-6995-7
  • Series Print ISSN 0066-5452
  • About this book