Professor Keith Promislow

Course Pages

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Spectral and Dynamical Stability of Nonlinear Waves

Ebook Available
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This book unifies the dynamical systems and functional analysis approaches to the linear and nonlinear stability of waves. It synthesizes the 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. Many exercises highlighting the main ideas are woven into the exposition. The book is intended for first or second year graduate students in mathematics, or those with equivalent mathematical maturity. 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 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.

Table of Contents
Ch.1 Introduction 1-4
Ch.2 Background Material and Notation 5-37
Ch.3 Essential and Absolute Spectra 39-74
Ch.4 Asymptotic Stability of Waves in Dissipative Systems 75-115
Ch.5 Orbital Stability of Waves in Hamiltonian Systems 117-157
Ch.6 Point Spectrum: Reduction to Finite-Rank Eigenvalue 159-175
Ch.7 Point Spectrum: Linear Hamiltonian Systems 177-213
Ch.8 The Evans Function for Boundary-Value Problems 215-247
Ch.9 The Evans Function for Sturm-Liouville Operators on the Real Line 248-303
Ch.10 The Evans Function for n'th-order Operators on the Real Line 305-344