\chapter{Introduction}
\label{chap1}
%
%\input{Introduction}

%Fronts and Pulses in nonlinear PDE, Periodic solutions
%in nonlinear PDE. Examples of nonlinear PDE, examples of
%front, pulse and periodic.

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 that in many ways placed 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, Jones, and Gardner %AU: Should this be Alexander et al. And not list all three authors, and should it have reference citation for ?[11]??% who recast the Evans function in a dynamical
systems language. The subsequent %It is the authors opinion that
%This surge of recent activity has brought the stability theory of nonlinear waves to the point where it can benefit from a unified treatment.
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.
%This unified framework is presented both in an
%abstract setting, and in the particular, with many aspects of the theory first motivated by detailed examples.
%in which the principles of functional analysis and dynamical systems are applied to the study of linear and nonlinear PDEs.
% within the realm of active knowledge.

Many graduate students in applied mathematics have been exposed to the key
ideas of dynamical systems and functional analysis by the end of their first
year; however, these fields are typically presented as unrelated. Within the
context of nonlinear waves, the goal of this book is to show how the tools of
dynamical systems provide a rich illumination of the abstract ideas of
functional analysis. However, the simultaneous application of these two
fields requires some sophistication. Our approach is to motivate the abstract
framework by first working through detailed examples that serve to
illustrate the moving parts, with the broader framework subsequently
presented as a generalization of a familiar process.
% Indeed, a key step in the mastery of a body of knowledge is the transition from the passive to the active phase of learning.
%Being familiar with a set of ideas is fundamentally different than knowing how and when, and most importantly why, to apply them. Watching
%a skilled metal smith at work, one is quickly impressed with the huge variety of
%hammers, each with a slightly different weight, head size and shape, and hardness;
%a significant part of the art of metal smithing is the map from the metal in hand
%to the hammer on the shelf.
%Our approach builds up an abstract framework which is heavily motivated through concrete examples.
As much as possible, we have emphasized the structure of the framework:
showing that stability and bifurcation can be understood through the
dynamical systems which characterize the underlying equilibria. We have
avoided the use of exact solutions of ordinary differential equations (ODEs),
% as we feel that for many students having an exact form,
%e.g. $\sech(x)$,
%serves as a crutch which encourages them to turn off the analytic mental machinery and focus instead on the more familiar machinations
%of first semester calculus.
whose detailed calculus tends to obfuscate the structure behind the ideas.

Towards this goal, \autoref{ch:background} provides a summary of background
material; essentially, what is assumed of a mathematics graduate
student who has completed a year of graduate PDE and functional analysis. We
understand that this knowledge may be incomplete and therefore we have made an effort to
integrate its application into the later chapters in a self-contained manner.
In many ways \autoref{ch:spectra} is the beginning of the book, applying
dynamical systems ideas to illuminate a fundamental result of functional
analysis: the Fredholm classification of linear differential operators on the
line. The Fredholm alternative is the underpinning of many existence and
bifurcation results, and its proof through dynamical systems techniques is
instructive. This approach places the solvability question for a ``linear
system of differential equations subject to boundary conditions'' inside of
the bigger box of ``all flows generated by the equivalent dynamical system.''
The task is to classify all solutions of the dynamical system, and then ask
which satisfy the boundary conditions. This recalls the classical
construction of a Green's function on a bounded domain, but the extension of
the ideas to
 the unbounded domain, particularly the classification step, leads naturally to the idea of the Evans function, and
 provides a concrete formulation of the essential spectrum of important classes of linear differential operators.

After \autoref{ch:spectra}, the flow of the book branches, leading either to a study of nonlinear stability and bifurcation
 via functional analytic techniques in Chaps.\,\ref{ch:nondyn}--\ref{ch:hseval}, or directly to the Evans function and the key
 applications of eigenvalue perturbation within the essential spectrum in Chaps.\,\ref{ch:evansbv}--\ref{ch:evansNorderRL}.
 The functional analytic tract is inspired qualitatively by the work of Benjamin, and more quantitatively by
 the work of Grillakis et al.%COMP: Add reference citations for 109 and 110%, who provided the first comprehensive treatment of the
 stability of Hamiltonian systems in the presence of symmetry. Our approach introduces the unifying idea
 of a constrained operator and a quantitative measure of the impact of linear constraint upon
 eigenvalue count. The constrained operator approach naturally abuts the Hamiltonian index theory, and plays a central role
 in the stability of critical points of Hamiltonian systems. Chapter~\ref{ch:nondyn}
presents an overview the properties of $\calC^0$ and analytic semigroups,
with attention to the role of symmetries in generating a point spectrum, and
the idea that spectral stability in many situations implies orbital stability
of a manifold of equilibria. In \autoref{ch:hamsys} we adapt these ideas to
the Hamiltonian framework, using an adaptation of Benjamin's proof of the
orbital stability of the solitary solution of the KdV equation to illustrate
the idea of a constrained operator within a concrete framework. The framework
is substantially generalized in \autoref{s:hamsys}, and is carried forward
to Chaps.\,\ref{ch:ls} and \ref{ch:hseval} with slight modification.
In \autoref{ch:ls} we present the formal perturbation structure of regular
spectra of linear operators, with concrete examples that illustrate the
major stability results of Chaps.\,\ref{ch:nondyn} and \ref{ch:hamsys}.
Chapter~\ref{ch:hseval} gives a detailed analysis of index theory for Hamiltonian
systems, introducing the Hamiltonian--Krein index, which enumerates the
potentially destabilizing point spectrum, and the Krein signature. In
particular, we develop an instability criterion that is complementary to the
nonlinear stability results of \autoref{ch:hamsys}. We also discuss
applications to bifurcation of point spectra with nontrivial Jordan block
structure under both symmetry-breaking Hamiltonian perturbations and symmetry
-preserving non-Hamiltonian perturbations.

The second part of the book develops the Evans function, an analytic
function of the spectral parameter of an associated eigenvalue problem, with zeros that coincide
with the point spectrum of the operator when they are within the natural domain of the Evans function.
The Evans function has analytic extensions beyond its natural domain, generically some
distance into the essential spectrum. This extension plays a fundamental role in understanding bifurcations associated with point spectra which are
ejected from the essential spectrum of the linearized operator under perturbation.

The machinery required to develop the Evans function is nontrivial; we first
consider the simpler context of boundary-value problems on a finite domain
in \autoref{ch:evansbv}. %and for spatially periodic problems in \autoref{ch:evansper}.
In this context the Evans function resembles a classical Wronskian, being an entire function
of the spectral parameter. The issues of branch points, branch cuts, and Riemann surfaces are deferred
until \autoref{ch:evanssl} which addresses second-order Sturm--Liouville operators
on the line. This requires a re-examination of the issue addressed in \autoref{ch:spectra}: the
classification of all the solutions of a given dynamical system by their asymptotic behavior,
which evokes the Jost functions of quantum mechanics. Even for this restricted class of problem one
finds the full richness of the eigenvalue problems:
branch points and branch cuts of the Evans function, and its extension beyond the natural domain and
onto Riemann sheets. Within the context of second-order differential operators we address two
important problems: the detection of real eigenvalues associated with
instabilities, and the tracking of eigenvalues as they enter and leave the
branch cuts. We also discuss the connection of the absolute spectrum to the large domain limit of the bounded
domain problem for both separated and periodic boundary conditions.
Chapter~\ref{ch:evansNorderRL} addresses issues arising in the extension of the Evans function to higher-order
linear operators, and incorporates several substantial examples that display the breadth of possible applications.

%However, we have integrated concrete examples with the abstraction, and in many places prepare the
%development of the abstract theory by first working through an illustrative example.

There are many relevant topics that are not
touched upon in this book. All issues associated with multiple scales and
(geometric) singular perturbation theory have been set aside.
Eigenvalue problems can also be considered through a topological version of the
Evans function, e.g., see \citep{kapitula:eas96,bose:sot95,rubin:sba99,gardner:two89}, or by
decomposing the Evans function into the product of meromorphic functions, each of which is associated with a
distinct time scale, e.g., see \citep{doelman:asi02,doelman:sao98}. The Green's function approach to
nonlinear stability, pioneered by Howard and Zumbrun, %\cite{},
is a natural outgrowth of the dynamical systems approach.
Another natural extension is to
%the orbital stability of manifolds which are only approximately invariant, but whose tangent plane is well approximated by the
%spectral space associated to the small eigenvalues of the associated linearized operator. These include symmetry breaking
%perturbations which generate a slow drift along the symmetry manifold, such as arises in
weak and semi-strong interaction regimes of multipulse solutions e.g., see
 % FIX-ME Shin-Eichiro Ei
\citep{sandstede:eas97,sandstede:som98,sandstede:guf00,yew:iom00,promislow:arm02,doelman:nas07}.
The most obvious limitation is that we have restricted ourselves to systems
in one space dimension; however, these and other extensions are the domain of
current research and do not seem ready for a unified treatment. The
literature in the field of nonlinear waves is vast, and we have made an
attempt to reference relevant papers at the end of each section; however, our
effort is of necessity incomplete and we apologize in advance to the many
people whose work we have not fully cited.

The content of the book has been used in a course for second-year graduate
students at Michigan State University. It could also naturally be used to
form two possible one-semester courses. The first course might emphasize the
functional analytic approach, and include Chaps.\,\ref{ch:spectra}--\ref{ch:hseval}. The second could
address the Evans function and dynamical systems approach, using Chaps.\,\ref{ch:spectra}
and \ref{ch:evansbv}--\ref{ch:evansNorderRL}. Both routes would be supplemented with material from Chap.\,\ref{ch:background} as
needed, and of course from the instructor's own personal repertoire of
examples.

% FIX-ME this belongs in the open remarks to Part II of the book.
%In the final four chapters we study a function, the \textsl{Evans function},
%which is analytic in the spectral parameter, and which has the property that
%it is zero if and only if there is an eigenvalue. The Evans function is
%basically a Wronskian for the linear ODE associated with the eigenvalue
%problem. In the mathematical physics community the Evans function is often
%called the \textsl{transmission coefficient} (see \citet{kapitula:ear04} and
%the references therein). The Evans function was introduced to the
%mathematical biology community in
%\citep{evans:naeI72,evans:naeII72,evans:naeIII72,evans:nae75}, and then
%brought to the wider mathematical community in
%\citep{jones:stw84,gardner:sis91,pego:eis92,pego:efm93}. Since that time it
%has undergone a variety of reformulations, extensions, and refinements, e.g.,
%see
%\citep{bridges:hda99,bridges:tse01,gardner:tgl98,kapitula:tef04,kapitula:sob98}.

%interest has come from areas as diverse as mathematical biology, physics,
%engineering, chemistry, etc. Accompanying this growth in (wo)manpower has
%been a significant development and maturation of the mathematical theory used
%to study and understand the dynamics. There are literally thousands of
%research articles in hundreds of journals on the subject.

%It is now the case that the theory used in cutting-edge research is
%well-beyond what a mathematically trained student will see in the first
%couple of years of graduate studies. This gap between formal study and the
%current theory presents a major hurdle for those students and researchers who
%are interested in using the mathematical tools and results in their own work.
%This book - lecture notes, really - is an attempt to bridge that gap.

%In this book we focus on one particular path that could be taken from the
%material normally presented in the first two years of graduate study in
%mathematics to being able to read and digest current journal articles. We do
%not attempt to present a comprehensive picture of the current
%state-of-the-art of the mathematical theory. Indeed, that would, in our
%opinion, make the book far too daunting to be an introductory text to the
%material. On the other hand, we have often chosen our applications of the
%theory to be problems which have been studied in the literature within the
%last twenty years. We believe that there is enough material in the book for a
%two-semester class. However, the individual chapters are written in such a
%manner that a coherent one-semester class could be taught from the material
%presented in the subsequent chapters.

%The second chapter is the cornerstone of the book. Herein we discuss the
%types of spectra associated with linear partial differential operators on the
%line. All of the subsequent chapters assume that the interested reader has
%digested this material. We set the notation for the book here, especially
%with respect to descriptive terms of the various types of spectra. We try to
%be consistent with the terminology currently in use; however, when we feel it
%is appropriate we introduce nonstandard terminology that we feel is a better
%descriptor. When we do so, we alert the reader that in the literature there
%is a different way of referring to the object. For example, in this book we
%call the eigenvalues of matrices \textsl{matrix eigenvalues}, whereas in the
%literature they are often called \textsl{spatial eigenvalues}. The remaining
%seven chapters serve, in some sense, as a sequence of special topics.

%In the third chapter we discuss how knowledge of the spectrum of the linear
%operator can under the correct assumptions provide provide definitive
%information about the stability of the nonlinear wave. We provide a sketch of
%the proof, and then focus on applying the theory to two substantive examples.
%In the fourth chapter we consider a class of problems for which it naively
%appears the linear theory is not definitive: Hamiltonian systems. Instead of
%first stating the general theory, and then using it in examples, we first
%consider a substantive example in detail: the question of stability of waves
%to the generalized Korteweg-de Vries equation. We focus on interpreting the
%problem from a geometric perspective, and we present a thorough sketch of the
%argument used to show that under a certain condition the wave is a local
%minimizer of the constrained energy, which in turn implies that it is
%orbitally stable. Afterwards, we show how the argument can be generalized to
%a large class of systems. In conclusion, the third and fourth chapters are
%the material on the nonlinear theory associated with (asymptotic) orbital
%stability.

%The remainder of the book is concerned with the study of the point spectrum
%of the operator. In the fifth chapter we consider the problem of perturbing
%the linear operator, and then looking at how the known eigenvalue(s) for the
%unperturbed problem move under the perturbation. We present the
%Lyapunov-Schmidt reduction theory for two cases which often appear in
%applications, and illustrate the theory with several substantive examples. We
%finish the chapter by thoroughly studying the perturbation problem in the
%context of Hamiltonian systems. The structure of the eigenvalue problem for
%these systems allows for us to make a definitive relationship between the
%nature of the critical point on the energy surface (minimum or saddle-point)
%and the point spectrum of the linearized operator. This perspective also
%allows us to consider the perturbed problem from a geometric perspective.

%