Dynamics of Codimension-One Defects

• P. Coullet
• C. Elphick
• D. Repaux
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 41)

Abstract

Pattern formation in nonequilibrium systems is associated with instabilities which break spatio-temporal symmetries [1], [2], The dynamics of a physical system is generally described by a set of partial differential equations of the type
$$\frac{dU}{dt}={F}_{\lbrace{\lambda}\rbrace}({U};{\nabla})$$
(1)
where U(r, t) is a state vector, V stands for the gradient operator, r represents a position vector and {λ} is a set of parameters. We assume that this system possesses a stationary solution denoted U*(r). A perturbation u(r, t) to this solution obeys the following equation:
$$\frac{\partial U}{\partial t} = {L}_{\{\lambda\}}( \nabla ) u + N(u),$$
(2)
where L{λ}(∇) is a linear operator defined as $$L=\frac{\delta F}{\delta U}\vert_{U^*}$$ stands for the non-linear terms. The stability of the stationary solution is determined by the spectrum of the linear operator L {λ} . We also assume that, for some {λ} = {λc} in the parameter space, the linear operator possesses an eigenvalue whose real part vanishes, while the others have a negative real part. Near the instability threshold, the perturbation of the stationary solution can be expressed as:
$$u(r, t) = A(r, t)\phi(r)+cc+V$$
(3)
where A(r,t) is the amplitude of the unstable mode, Φ(r) is the corresponding eigenvector which keeps track of the broken symmetries, and V contains the higher order contributions coming from the non-linear part of the unstable mode.

Keywords

Chaotic Behavior Unstable Mode Instability Threshold Ferromagnetic Transition High Order Contribution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
P.C. Hohenberg and J.S. Langer, J. Stat. Phys. 28, 193 (1982).
2. 2.
J.E. Wesireid and S. Zaleski, eds. Cellular Structures in Instabilities (Springer, New York, 1984).
3. 3.
G. Toulouse and M. Kleman, J.Phys.Lett. 37, 149 (1976).
4. 4.
For a review see N.D. Mermin, Rev. Mod.Phys. 51, 592 (1979).
5. 5.
P. Coullet, C. Elphick, L. Gil and J. Lega, Phys. Rev. Lett. 59, 884 (1987); and see L. Gil and J. Lega in these proceedings.
6. 6.
P. Coullet and C. Elphick, Phys. Lett. A121, 234 (1987).
7. 7.
J. Guckenheimer and P. Holmes, Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, Berlin, Heidelberg 1983), and references quoted therein.Google Scholar
8. 8.
K. Kawasaki and T. Ohta, Physica 116A, 573 (1982).
9. 9.
S. Aubry, in Solitons and Condensed Matter Physics. Springer Ser. Solid State Phys., Vol. 8, edited by A. R. Bishop and T. Schneider (Springer, Berlin, 1979).Google Scholar
10. 10.
P. Bak, Rep. Prog. Phys. 45, 587 (1982), and references quoted therein.
11. 11.
P. Coullet, C. Elphick and D. Repaux, Phys. Rev. Lett. 58, 431 (1987).
12. 12.
W.J. Firth, “Optical Memory and Spatial Chaos”, Preprint Glasgow University.Google Scholar
13. 13.
R. M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. 35, 1678 (1975).
14. 14.
A. E. Jacobs and B. Walker, Phys. Rev. 21, 4132 (1980).
15. 15.
P. Coullet, Phys. Rev. Lett. 56, 724 (1986).
16. 16.
P. Coullet and D. Repaux, Europhys. Lett., 3, 573 (1987).
17. 17.
M. Lowe and J. Gollub, Phys. Rev. A31, 3893 (1983).
18. 18.
See P. Manneville in Structures et Instabilités, C. Godreche eds, les Editions de la Physique, p. 423 (1986).Google Scholar
19. 19.
J. Wesfreid, Thesis, Universite de Paris-Sud Centre d’Orsay, p. 38 (1981).Google Scholar

Authors and Affiliations

• P. Coullet
• 1
• C. Elphick
• 1
• D. Repaux
• 1
1. 1.Laboratoire de Physique ThéoriqueUniversité de NiceNice CedexFrance