Spectral and dynamical stability of nonlinear waves
 Author/Creator
 Kapitula, Todd author.
 Language
 English.
 Publication
 New York, NY : Springer Verlag [2013]
 Physical description
 xiii, 361 pages : illustrations ; 24 cm.
 Series
 Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 185.
Access
Contributors
 Contributor
 Promislow, Keith, 1964 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 345357) and index.
 Contents

 Background material and notation
 Linear systems of ordinary differential equations
 Constant matrices : the matrix exponential
 Constant matrices : invariant subspaces and estimates on solutions
 Periodic matrices : floquet theory
 General matrices and exponential dichotomies
 Elements of functional analysis
 Basic Sobolev spaces
 Bounded and closed operators
 Variational derivatives
 Resolvent and spectrum
 Adjoint and Fredholm operators
 The point spectrum : SturmLiouville theory
 SturmLiouville operators on a bounded domain
 SturmLiouville operators on the real line
 Examples
 Additional reading
 Essential and absolute spectra
 The essential spectrum : fronts and pulses
 Examples
 The absolute spectrum
 Examples
 Absolute spectrum and the large domain limit
 The essential spectrum : periodic coefficients
 Example : Hill's equation
 Additional reading
 Asymptotic stability of waves in dissipative systems
 Linear dynamics
 Systems with symmetries
 Nonlinear dynamics
 Example : scalar viscous conservation law
 Example : nonlinear Schrödingertype equations
 Additional reading
 Orbital stability of waves in Hamiltonian systems
 Finitedimensional systems
 Infinitedimensional Hamiltonian systems with symmetry
 The generalized Kortewegde Vries equation
 General orbital stability result
 Eigenvalues of constrained selfadjoint operators
 Additional reading
 Point spectrum : reduction to finiterank eigenvalue problems
 Perturbation of an algebraically simple eigenvalue
 Example : parametrically forced GinzburgLandau equation
 Example : spatially periodic waves of gKdV
 Perturbation of a geometrically simple eigenvalue
 Point spectrum : linear Hamiltonian systems
 The Krein signature and the HamiltonianKrein index
 A finitedimensional version of theorem 7.1.5
 Krein signature and bifurcation
 The JonesGrillakis instability index
 Symmetrybreaking perturbations
 Hamiltonian perturbation
 NonHamiltonian perturbations
 Additional reading
 The Evans function for boundaryvalue problems
 SturmLiouville operators
 Higherorder operators
 Rigorous multiplicity proof : mg ... = 1*
 Rigorous multiplicity proof : mg ... > ̲2*
 Secondorder systems
 The Evans function for periodic problems
 Application : spectral properties
 Additional peading
 The Evans function for SturmLiouville operators on the real line
 The wholeline eigenvalue problem
 Spectral projections and the Jost solutions
 The Evans function
 Example : squarewell potential
 Example : reflectionless potential
 Application : the orientation index
 Application : edge bifurcations
 The ... = 0 problem
 Calculation of ...
 Calculation of ...
 Application : eigenvalue problems on large intervals with separated boundary conditions
 Application : eigenvalue problems for periodic problems with large spatial period
 Additional reading
 The Evans function for nthorder operators on the real line
 The Jost matrices
 The Evans function
 Application : the orientation index
 Example : generalized kortewegde vries equation
 Example : parametrically gorced GinzburgLandau equation
 Application : edge bifurcations
 Example : the nonlinear Schrödinger equation
 Example : a perturbed Manakov equation
 Eigenvalue problems on large intervals : separated boundary conditions
 Eigenvalue problems : periodic coefficients with a large spatial period
 Additional reading
 References
 Index.
 Publisher's Summary
 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.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2013
 Responsibility
 Todd Kapitula, Keith Promislow.
 Series
 Applied mathematical sciences, 00665452 ; volume 185
 Note
 Also issued online.
 ISBN
 9781461469940
 1461469945