Spectral and Dynamical Stability of Nonlinear Waves

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.

• Evans function
• Hamiltonian systems
• Lyapunov-Schmidt reductions
• Nonlinear Waves
• Spectral Theory
• partial differential equations

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)

• , Department of Mathematics and Statistics, Calvin College, Grand Rapids, USA

  Todd Kapitula

• , Department of Mathematics, Michigan State University, East Lansing, USA

  Keith Promislow

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.

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