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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.

Inhaltsverzeichnis

Chapter 1. Introduction

Abstract
The stability of nonlinear waves has a distinguished history and an abundance of richly structured yet accessible examples, which makes it not only an important subject but also an ideal training ground for the study of linear and nonlinear partial differential equations (PDEs). While the ”modern” approach to the stability of nonlinear waves can be traced back to the key papers of Joseph Boussinesq in the 1870s, the field has experienced tremendous growth over the past 30 years. Some of the growth was stimulated by T. B. Benjamin’s 1972 paper, ”The stability of solitary waves,” which presented a treatment of the Korteweq–de Vries equation in that served to place meat on the bones of Boussinesq’s ideas. A more recent avenue of growth stems from the development of dynamical systems ideas, which provide a rich complement to the functional analytic approach. In many ways these developments were stimulated by the pioneering work of Alexander et al. who recast the Evans function in a dynamical systems language. The subsequent synergy between dynamical systems and functional analysis has yielded a burst of activity and produced a unified framework for the study of stability and bifurcation in nonlinear waves.
Todd Kapitula, Keith Promislow

Chapter 2. Background Material and Notation

Abstract
This chapter provides an overview of the background material assumed in the remainder of the book. We state major results and provide sketches of the less technical proofs, particularly where the ideas presented are instrumental in subsequent constructions. The first topic is the theory of linear systems of ordinary differential equations (ODEs). Much of this material is standard for first-year graduate courses in ODEs, such as presented in Hartman [114], Perko [234]. This is followed by a review of the basic theory of functional analysis as applied to linear partial differential equations; further details can be found in the standard references Evans [81], Kato [162]. We finish the general overview by discussing the point spectrum in the context of the Sturm–Liouville theory for second-order operators. These operators have a one-to-one relationship between the ordering of the eigenvalues and the number of zeros for the associated eigenfunctions, which is extremely useful in applications.
Todd Kapitula, Keith Promislow

Chapter 3. Essential and Absolute Spectra

Abstract
The goal of this chapter is the characterization of the essential spectrum and Fredholm indices of two classes of linear differential operators on unbounded domains. The first class is comprised of nth-order differential operators with spatially varying coefficients that tend at an exponential rate to constant values at $$\pm \infty$$.
Todd Kapitula, Keith Promislow

Chapter 4. Asymptotic Stability of Waves in Dissipative Systems

Abstract
A key motivation for investigating the spectrum of linear operators is to understand the stability of equilibria of nonlinear evolution equations, as well as to describe the flow in a neighborhood of manifolds of approximate equilibria.
Todd Kapitula, Keith Promislow

Chapter 5. Orbital Stability of Waves in Hamiltonian Systems

Abstract
Hamiltonian systems arise in a myriad of applications where damping can be neglected, from the motion of celestial bodies, to the spinning of rigid tops, to interactions of particles in molecular systems. They are also imbued with a rich structure that arises from the conservation of the underlying energy, the Hamiltonian, as well as other quantities such as mass and momentum. In this chapter we present a theory for the nonlinear stability of generalized traveling-wave solutions of Hamiltonian systems. This field has a long history, starting with a conjecture of Boussinesq, dating to 1872 [39], in which he suggested the constraint structure could be used to understand the stability of the critical points of the Hamiltonian.
Todd Kapitula, Keith Promislow

Chapter 6. Point Spectrum: Reduction to Finite-Rank Eigenvalue Problems

Abstract
The word bifurcation refers to changes in the number and stability of equilibria supported by a governing system as its parameters are varied. The classical bifurcation problem begins with an analysis of the point spectrum of the linearized operator associated with the equilibria under investigation. In this chapter we investigate finite-rank bifurcations for which a finite number of point eigenvalues cross the imaginary axis, either transversely or more degenerately, as the system parameters are varied. In particular, we derive the perturbative motion of such point spectra. This analysis is most informative in those cases for which the associated linearized operator initially has purely imaginary eigenvalues, and a small change in parameters moves the eigenvalues decisively off the imaginary axis.
Todd Kapitula, Keith Promislow

Chapter 7. Point Spectrum: Linear Hamiltonian Systems

Abstract
Hamiltonian systems are about balance, with the energy and other invariants preserved under the flow. For a spatially localized critical point of a Hamiltonian system, the balance is reflected in the symmetry of the spectrum, which typically pins the essential spectrum to the imaginary axis in unweighted spaces. The mechanism for bifurcation in Hamiltonian systems thus falls upon the point spectrum.
Todd Kapitula, Keith Promislow

Chapter 8. The Evans Function for Boundary-Value Problems

Abstract
Previously we gathered information about a point spectrum either perturbatively, as in Chapter 6, or in cases where the linear operator has special structure, as arises from symmetries (Chapter 4.2) and in Hamiltonian systems (Chapter 7). In this chapter we construct the Evans function, an analytic function of the spectral parameter with the property that its zeros correspond to eigenvalues with the order of the zero equal to the algebraic multiplicity of the eigenvalue.
Todd Kapitula, Keith Promislow

Chapter 9. The Evans Function for Sturm–Liouville Operators on the Real Line

Abstract
In this chapter we construct the Evans function for exponentially asymptotic differential operators on unbounded domains. While several key elements of the construction naturally carry over from the bounded domain construction, important new subtleties arise. To focus on these key issues, we restrict our attention to second-order Sturm–Liouville operators. The extension to higher-order linear operators in addressed in Chapter 10.
Todd Kapitula, Keith Promislow

Chapter 10. The Evans Function for nth-Order Operators on the Real Line

Abstract
The primary goal of this chapter is the construction of the Evans function for eigenvalue problems associated with nth-order, exponentially asymptotic linear operators acting on L2(R). The construction, through the Jost solutions, is distinguished from the construction for second-order operators by the fact that the matrix eigenvalues and associated eigenvectors for the nth-order problem may not be analytic in the natural domain of the Evans function. Moreover, while it is relatively easy to determine the essential spectrum for these problems, the matrix eigenvalues and the absolute spectrum do not generally have an explicit representation. We sidestep these issues via an analytic extension of the stable and unstable spaces of the asymptotic matrix which leads to the construction of Jost matrices.
Todd Kapitula, Keith Promislow

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