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Spectral and dynamical stability of nonlinear waves / Todd Kapitula, Keith Promislow.

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Author/Creator:
Kapitula, Todd author.
Language:
English.
Publication date:
2013
Publication:
New York, NY : Springer Verlag [2013]
Format:
  • Book
  • xiii, 361 pages : illustrations ; 24 cm.
Bibliography:
Includes bibliographical references (pages 345-357) 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 : Sturm-Liouville theory
  • Sturm-Liouville operators on a bounded domain
  • Sturm-Liouville 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 Schrödinger-type equations
  • Additional reading
  • Orbital stability of waves in Hamiltonian systems
  • Finite-dimensional systems
  • Infinite-dimensional Hamiltonian systems with symmetry
  • The generalized Korteweg-de Vries equation
  • General orbital stability result
  • Eigenvalues of constrained self-adjoint operators
  • Additional reading
  • Point spectrum : reduction to finite-rank eigenvalue problems
  • Perturbation of an algebraically simple eigenvalue
  • Example : parametrically forced Ginzburg-Landau equation
  • Example : spatially periodic waves of gKdV
  • Perturbation of a geometrically simple eigenvalue
  • Point spectrum : linear Hamiltonian systems
  • The Krein signature and the Hamiltonian-Krein index
  • A finite-dimensional version of theorem 7.1.5
  • Krein signature and bifurcation
  • The Jones-Grillakis instability index
  • Symmetry-breaking perturbations
  • Hamiltonian perturbation
  • Non-Hamiltonian perturbations
  • Additional reading
  • The Evans function for boundary-value problems
  • Sturm-Liouville operators
  • Higher-order operators
  • Rigorous multiplicity proof : mg ... = 1*
  • Rigorous multiplicity proof : mg ... > ̲2*
  • Second-order systems
  • The Evans function for periodic problems
  • Application : spectral properties
  • Additional peading
  • The Evans function for Sturm-Liouville operators on the real line
  • The whole-line eigenvalue problem
  • Spectral projections and the Jost solutions
  • The Evans function
  • Example : square-well 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 nth-order operators on the real line
  • The Jost matrices
  • The Evans function
  • Application : the orientation index
  • Example : generalized korteweg-de vries equation
  • Example : parametrically gorced Ginzburg-Landau 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.
Note:
Also issued online.
Contributor:
Promislow, Keith, 1964- author.
Series:
Applied mathematical sciences, 0066-5452 ; volume 185
Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 185.
Subjects:
ISBN:
9781461469940
1461469945

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