Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals

Beyn W-J, Lorenz J (1999)
Numerical Functional Analysis and Optimization 20(3-4): 201-244.

Zeitschriftenaufsatz | Veröffentlicht| Englisch
 
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Abstract / Bemerkung
If a traveling wave is stable or unstable depends essentially on the spectrum of a differential operator P obtained by linearization. We investigate how spectral properties of the all-line operator P are related to eigenvalues of finite-interval BVPs Pu(x) = su(x), x - less than or equal to x less than or equal to x(+), Ru = 0. Here R is a linear boundary operator; for which we will derive determinant conditions, and the x-interval is assumed to be sufficiently large. Under suitable assumptions, we show (a) resolvent estimates for large s; (b) if s is in the resolvent of the all-line operator P, then s is also in the resolvent for finite-interval BVPs; (c) eigenvalues of P lead to approximating eigenvalues on finite intervals. These results allow to study the stability question for traveling waves by investigating eigenvalues of finite-interval problems. We give applications to the FitzHugh-Nagumo system with small diffusion and to the complex Ginzburg-Landau equations.
Stichworte
exponential dichotomies; eigenvalues; finite boundary value problems; traveling waves; stability
Erscheinungsjahr
1999
Zeitschriftentitel
Numerical Functional Analysis and Optimization
Band
20
Ausgabe
3-4
Seite(n)
201-244
ISSN
0163-0563
Page URI
https://pub.uni-bielefeld.de/record/1622883

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Beyn W-J, Lorenz J. Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals. Numerical Functional Analysis and Optimization. 1999;20(3-4):201-244.
Beyn, W. - J., & Lorenz, J. (1999). Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals. Numerical Functional Analysis and Optimization, 20(3-4), 201-244. doi:10.1080/01630569908816889
Beyn, W. - J., and Lorenz, J. (1999). Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals. Numerical Functional Analysis and Optimization 20, 201-244.
Beyn, W.-J., & Lorenz, J., 1999. Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals. Numerical Functional Analysis and Optimization, 20(3-4), p 201-244.
W.-J. Beyn and J. Lorenz, “Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals”, Numerical Functional Analysis and Optimization, vol. 20, 1999, pp. 201-244.
Beyn, W.-J., Lorenz, J.: Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals. Numerical Functional Analysis and Optimization. 20, 201-244 (1999).
Beyn, Wolf-Jürgen, and Lorenz, Jens. “Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals”. Numerical Functional Analysis and Optimization 20.3-4 (1999): 201-244.