"Spectral and Dynamical Stability of Nonlinear Waves" by Todd Kapitula and Keith Promislow

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 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.

Contents
Introduction
Background Material and Notation
Essential and Absolute Spectra
Asymptotic Stability of Waves in Dissipative Systems
Orbital Stability of Waves in Hamiltonian Systems
Point Spectrum: Reduction to Finite-Rank Eigenvalue Problems
Point Spectrum: Linear Hamiltonian Systems
The Evans Function for Boundary-Value Problems
The Evans Function for Sturm–Liouville Operators on the Real Line
The Evans Function for nth-Order Operators on the Real Line