Semi-Fredholm operators and hyperbolic problems

2006 ◽  
pp. 249-264
Author(s):  
Mario Martelli
Keyword(s):  
Hyperbolic Problems ◽  
Fredholm Operators

Global Perturbation of Nonlinear Eigenvalues

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Perturbation Theory ◽  
Spectral Parameter ◽  
Classical Theory ◽  
General Setting ◽  
Perturbation Parameter ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Index Zero ◽  
Finite Dimensional ◽  
Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

Finite volume methods, unstructured meshes and strict stability for hyperbolic problems

2003 ◽  
Vol 45 (4) ◽  
pp. 453-473 ◽  
Author(s):  
Jan Nordström ◽  
Karl Forsberg ◽  
Carl Adamsson ◽  
Peter Eliasson
Keyword(s):  
Finite Volume ◽  
Unstructured Meshes ◽  
Finite Volume Methods ◽  
Hyperbolic Problems ◽  
Strict Stability

Economical finite rank perturbations of semi-Fredholm operators

1988 ◽  
Vol 198 (3) ◽  
pp. 431-434 ◽  
Author(s):  
M�che�l � Searc�id
Keyword(s):  
Finite Rank ◽  
Fredholm Operators

NONLINEAR ASYMPTOTICS FOR HYPERBOLIC INTERNAL WAVES OF SMALL WIDTH

2006 ◽  
Vol 03 (02) ◽  
pp. 269-295 ◽  
Author(s):  
OLIVIER GUES ◽  
JEFFREY RAUCH
Keyword(s):  
Internal Waves ◽  
Transition Layer ◽  
Layer Structure ◽  
Piecewise Smooth ◽  
Transmission Problem ◽  
Internal Layer ◽  
Smooth Transition ◽  
Hyperbolic Problems ◽  
Smooth Solutions ◽  
Small Width

Semilinear hyperbolic problems with source terms piecewise smooth and discontinuous across characteristic surfaces yield similarly piecewise smooth solutions. If the discontinuous source is replaced with a smooth transition layer, the discontinuity of the solution is replaced by a smooth internal layer. In this paper we describe how the layer structure of the solution can be computed from the layer structure of the source. The key idea is to use a transmission problem strategy for the problem with the smooth internal layer. That leads to an anastz different from the obvious candidates. The obvious candidates lead to overdetermined equations for correctors. With the transmission problem strategy we compute infinitely accurate expansions.

scholarly journals The validity of modulation equations for extended systems with cubic nonlinearities

1992 ◽  
Vol 122 (1-2) ◽  
pp. 85-91 ◽  
Author(s):  
Pius Kirrmann ◽  
Guido Schneider ◽  
Alexander Mielke
Keyword(s):  
Gordon Equation ◽  
Landau Equation ◽  
Hyperbolic Problems ◽  
Sine Gordon Equation ◽  
Ginzburg Landau ◽  
Modulation Equation ◽  
Modulation Equations ◽  
Quadratic Interaction ◽  
Long Time ◽  
Dissipative Case

SynopsisModulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems with no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.

Fredholm operators on locally compact spaces

1984 ◽  
Vol 138 (1) ◽  
pp. 191-210
Author(s):  
N. Teleman
Keyword(s):  
Fredholm Operators ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Spaces

Sharp Finite Speed for Hyperbolic Problems Well Posed in Gevrey Classes

2011 ◽  
Vol 36 (1) ◽  
pp. 1-9 ◽  
Author(s):  
F. Colombini ◽  
J. Rauch
Keyword(s):  
Hyperbolic Problems ◽  
Finite Speed ◽  
Gevrey Classes ◽  
Well Posed

On a new topology in the space of Fredholm operators

1989 ◽  
Vol 7 (2) ◽  
pp. 93-106 ◽  
Author(s):  
Anvar Irmatov
Keyword(s):  
Fredholm Operators

Minisymposium: Advanced Numerical Methods for Hyperbolic Problems

2017 ◽  
pp. 45-46
Author(s):  
Giovanni Russo ◽  
Sebastiano Boscarino
Keyword(s):  
Numerical Methods ◽  
Hyperbolic Problems

scholarly journals The instability of non-semi-Fredholm operators under compact perturbations

1982 ◽  
Vol 87 (2) ◽  
pp. 632-638 ◽  
Author(s):  
Richard Bouldin
Keyword(s):  
Fredholm Operators ◽  
Compact Perturbations
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