1 Introduction ................................................. 1
2 Background Material and Notation ............................. 5
2.1 Linear Systems of Ordinary Differential Equations ....... 5
2.1.1 Constant Matrices: The Matrix Exponential ........ 9
2.1.2 Constant Matrices: Invariant Subspaces and
Estimates on Solutions .......................... 12
2.1.3 Periodic Matrices: Floquet Theory ............... 15
2.1.4 General Matrices and Exponential Dichotomies .... 19
2.2 Elements of Functional Analysis ........................ 20
2.2.1 Basic Sobolev Spaces ............................ 20
2.2.2 Bounded and Closed Operators .................... 23
2.2.3 Variational Derivatives ......................... 24
2.2.4 Resolvent and Spectrum .......................... 25
2.2.5 Adjoint and Fredholm Operators .................. 27
2.3 The Point Spectrum: Sturm-Liouville Theory ............. 30
2.3.1 Sturm-Liouville Operators on a Bounded Domain ... 30
2.3.2 Sturm-Liouville Operators on the Real Line ...... 33
2.3.3 Examples ........................................ 33
2.4 Additional Reading ..................................... 37
3 Essential and Absolute Spectra .............................. 39
3.1 The Essential Spectrum: Fronts and Pulses .............. 39
3.1.1 Examples ........................................ 52
3.2 The Absolute Spectrum .................................. 60
3.2.1 Examples ........................................ 64
3.2.2 Absolute Spectrum and the Large Domain Limit .... 65
3.3 The Essential Spectrum: Periodic Coefficients .......... 67
3.3.1 Example: Hill's Equation ........................ 70
3.4 Additional Reading ..................................... 74
4 Asymptotic Stability of Waves in Dissipative Systems ........ 75
4.1 Linear Dynamics ........................................ 77
4.2 Systems with Symmetries ................................ 86
4.3 Nonlinear Dynamics ..................................... 90
4.4 Example: Scalar Viscous Conservation Law ............... 99
4.5 Example: Nonlinear Schrцdinger-Type Equations ......... 107
4.6 Additional Reading .................................... 114
5 Orbital Stability of Waves in Hamiltonian Systems .......... 117
5.1 Finite-Dimensional Systems ............................ 118
5.2 Infinite-Dimensional Hamiltonian Systems with
Symmetry .............................................. 122
5.2.1 The Generalized Korteweg-de Vries Equation ..... 123
5.2.2 General Orbital Stability Result ............... 136
5.3 Eigenvalues of Constrained Self-Adjoint Operators ..... 148
5.4 Additional Reading .................................... 156
6 Point Spectrum: Reduction to Finite-Rank Eigenvalue
Problems ................................................... 159
6.1 Perturbation of an Algebraically Simple Eigenvalue .... 160
6.1.1 Example: Parametrically Forced Ginzburg-
Landau Equation ................................ 162
6.1.2 Example: Spatially Periodic Waves of gKdV ...... 165
6.2 Perturbation of a Geometrically Simple Eigenvalue ..... 173
7 Point Spectrum: Linear Hamiltonian Systems ................. 177
7.1 The Krein Signature and the Hamiltonian-Krein Index ... 179
7.1.1 A Finite-Dimensional Version of Theorem 7.1.5 .. 183
7.1.2 Krein Signature and Bifurcation ................ 187
7.1.3 The Jones-Grillakis Instability Index .......... 188
7.2 Symmetry-Breaking Perturbations ....................... 193
7.2.1 Hamiltonian Perturbation ....................... 193
7.2.2 Non-Hamiltonian Perturbations .................. 204
7.3 Additional Reading .................................... 212
8 The Evans Function for Boundary-Value Problems ............. 215
8.1 Sturm-Liouville Operators ............................. 215
8.2 Higher-Order Operators ................................ 225
8.2.1 Rigorous Multiplicity Proof: mg(λ0) = 1* ....... 232
8.2.2 Rigorous Multiplicity Proof: mg(λ0) ≥ 2* ....... 234
8.3 Second-Order Systems .................................. 237
8.4 The Evans Function for Periodic Problems .............. 240
8.4.1 Application: Spectral Properties ............... 243
8.5 Additional Reading .................................... 247
9 The Evans Function for Sturm-Liouville Operators on the
Real Line .................................................. 249
9.1 The Whole-Line Eigenvalue Problem ..................... 250
9.2 Spectral Projections and the Jost Solutions ........... 253
9.3 The Evans Function .................................... 262
9.3.1 Example: Square-Well Potential ................. 267
9.3.2 Example: Reflectionless Potential .............. 269
9.4 Application: The Orientation Index .................... 272
9.5 Application: Edge Bifurcations ........................ 276
9.5.1 The Є = 0 Problem .............................. 278
9.5.2 Calculation of ∂γE(0,0) ........................ 280
9.5.3 Calculation of ∂ЄE(0,0) ........................ 284
9.6 Application: Eigenvalue Problems on Large Intervals
with Separated Boundary Conditions .................... 291
9.7 Application: Eigenvalue Problems for Periodic
Problems with Large Spatial Period .................... 299
9.8 Additional Reading .................................... 303
10 The Evans Function for nth-Order Operators on the Real
Line ....................................................... 305
10.1 The Jost Matrices ..................................... 306
10.2 The Evans Function .................................... 314
10.3 Application: The Orientation Index .................... 316
10.3.1 Example: Generalized Korteweg-de Vries
Equation ....................................... 318
10.3.2 Example: Parametrically Forced Ginzburg-
Landau Equation ................................ 321
10.4 Application: Edge Bifurcations ........................ 326
10.4.1 Example: The Nonlinear Schrцdinger Equation .... 328
10.4.2 Example: A Perturbed Manakov Equation .......... 332
10.5 Eigenvalue Problems on Large Intervals: Separated
Boundary Conditions ................................... 338
10.6 Eigenvalue Problems: Periodic Coefficients with
a Large Spatial Period ................................ 341
10.7 Additional Reading ................................... 344
References ................................................. 345
Index ......................................................... 359
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